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Algebraic Invariants. By LEONARD E. DICKSON. $1.25 net. No. 15. Mortality Laws and Statistics. By ROBERT HENDERSON. $1.25 net. No. 16. Diophantine Analysis. By ROBERT D. CARMICHAEL. $1.25 net. No. 17. Ten British Mathematicians. By ALEX- ANDER MACFARLANE. $1.25 net. PUBLISHED BY JOHN WILEY & SONS, Inc., NEW YORK. CHAPMAN & HALL, Limited, LONDON MATHEMATICAL MONOGRAPHS. EDITED BY MANSFIELD MERRIMAN AND ROBERT S. WOODWARD. No. 9. DIFFERENTIAL EQUATIONS. BY W. WOOLSEY JOHNSON, PROFESSOR OF MATHEMATICS IN THE UNITED STATES NAVAL ACADEMY. FOURTH EDITION. FIRST THOUSAND. NEW YORK: JOHN WILEY & SONS. LONDON: CHAPMAN & HALL, LIMITED. 1906. COPYRIGHT, 1896, BY MANSFIELD MERRIMAN AND ROBERT S. WOODWARD UNDER THE TlTLE HIGHER MATHEMATICS. First Edition, September, 1896. Second Edition, January, 1898. Third Edition, August, 1900. Fourth Edition, January, 1906. FOHRT DRUMMOND, PRINTRR, NRW YORIC. EDITORS' PREFACE. THE volume called Higher Mathematics, the first edition of which was published in 1896, contained eleven chapters by eleven authors, each chapter being independent of the others, but all supposing the reader to have at least a mathematical training equivalent to that given in classical and engineering colleges. The publication of that volume is now discontinued and the chapters are issued in separate form. In these reissues it will generally be found that the monographs are enlarged by additional articles or appendices which either amplify the former presentation or record recent advances. This plan of publication has been arranged in order to meet the demand of teachers and the convenience of classes, but it is also thought that it may prove advantageous to readers in special lines of mathematical literature. It is the intention of the publishers and editors to add other monographs to the series from time to time, if the call for the same seems to warrant it. Among the topics which are under consideration are those of elliptic functions, the theory of num- bers, the group theory, the calculus of variations, and non- Euclidean geometry; possibly also monographs on branches of astronomy, mechanics, and mathematical physics may be included.. It is the hope of the editors that this form of publication may tend to promote mathematical study and research over a wider field than that which the former volume has occupied. December, 1905. AUTHOR'S PREFACE. IT is customary to divide the Infinitesimal Calculus, or Calcu- lus of Continuous Functions, into three parts, under the heads Differential Calculus, Integral Calculus, and Differential Equa- tions. The first corresponds, in the language of Newton, to the "direct method of tangents, " the other two to the " inverse method of tangents"; while the questions which come under this last head he further -divided into those involving the two fluxions and one fluent, and those involving the fluxions and both fluents. On account of the inverse character which thus attaches to the present subject, the differential equation must necessarily at first be viewed in connection with a "primitive," from which it might have been obtained by the direct process, and the solu- tion consists in the discovery, by tentative and more or less arti- ficial methods, of such a primitive, when it exists; that is to say, when it is expressible in the elementary functions which constitute the original field with which the Differential Calculus has to do. It is the nature of an inverse process to enlarge the field of its operations, and the present is no exception; but the adequate handling of the new functions with which the field is thus enlarged requires the introduction of the complex variable, and is beyond the scope of a work of this size. But the theory of the nature and meaning of a differential equation between real variables possesses a great deal of interest. To this part of the subject I have endeavored to give a full treat- ment by means of extensive use of graphic representations in AUTHOR S PREFACE. V rectangular coordinates. If we ask what it is that satisfies an ordinary differential equation of the first order, the answer must be certain sets of simultaneous values of x, y, and p. The geo- metrical representation of such a set is a point in a plane asso- ciated with a direction, so to speak, an infinitesimal stroke, and the "solution" consists of the grouping together of these strokes into curves of which they form elements. The treatment of singular solutions, following Cayley, and a comparison with the methods previously in use, illustrates the great utility of this point of view. Again, in partial differential equations, the set of simultaneous values of x, y, z, p, and q which satisfies an equation of the first order is represented by a point in space associated with the direc- tion of a plane, so to speak by a flake, and the mode in which these coalesce so as to form linear surface elements and con- tinuous surfaces throws light upon the nature of general and complete integrals and of the characteristics. The expeditious symbolic methods of integration applicable to some forms of linear equations, and the subject of development of integrals in convergent series, have been treated as fully as space would allow. Examples selected to illustrate the principles developed in each section will be found at its close, and a full index of subjects at the end of the volume. W. W. J. ANNAPOLIS, MD., December, 1905. CONTENTS. ART. i. EQUATIONS OF FIRST ORDER AND DEGREE Page i 2. GEOMETRICAL REPRESENTATION 3 3. PRIMITIVE OF A DIFFERENTIAL EQUATION 5 4. EXACT DIFFERENTLY EQUATIONS 6 5. HOMOGENEOUS EQUATIONS 9 6. THE LINEAR EQUATION 10 7. FIRST ORDER AND SECOND DEGREE 12 8. SINGULAR SOLUTIONS 15 9. SINGULAR SOLUTION FROM THE COMPLETE INTEGRAL . . . 18 10. SOLUTION BY DIFFERENTIATION 20 11. GEOMETRIC APPLICATIONS; TRAJECTORIES 23 12. SIMULTANEOUS DIFFERENTIAL EQUATIONS 25 13. EQUATIONS OF THE SECOND ORDER 28 14. THE Two FIRST INTEGRALS 31 15. LINEAR EQUATIONS 34 1 6. LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS 36 17. HOMOGENEOUS LINEAR EQUATIONS 40 18. SOLUTIONS IN INFINITE SERIES 42 19. SYSTEMS OF DIFFERENTIAL EQUATIONS 47 20. FIRST ORDER AND DEGREE WITH THREE VARIABLES ... .50 21. PARTIAL DIFFERENTIAL EQUATIONS OF FIRST ORDER AND DEGREE 53 22. COMPLETE AND GENERAL INTEGRALS 57 23. COMPLETE INTEGRAL FOR SPECIAL FORMS 60 24. PARTIAL EQUATIONS OF SECOND ORDER 63 25. LINEAR PARTIAL DIFFERENTIAL EQUATIONS 66 INDEX 72 DIFFERENTIAL EQUATIONS. ART. 1. EQUATIONS -OF FIRST ORDER AND DEGREE. In the Integral Calculus, supposing y to denote an unknown- function of the independent variable x, the derivative of y with respect to x is given in the form of a function of x, and it is required to find the value of y as a function of x. In other words, given an equation of the form g=/O), or dy = flx)dx t (I) of which the general solution is written in the form y = //(*X*, (2) it is the object of the Integral Calculus to reduce the expres- sion in the second member of equation (2) to the form of a known function of x. When such reduction is not possible, the equation serves to define a new function of x. In the extension of the processes of integration of which the following pages give a sketch the given expression for the derivative may involve not only x, but the unknown function y ; or, to write the equation in a form analogous to equation (i), it may be Mdx + Ndy = o, (3) in which J/and TV are functions of x and y. This equation is in fact the general form of the differential equation of the first order and degree ; either variable being taken as the independ- ent variable, it gives the first derivative of the other variable 2 DIFFERENTIAL EQUATIONS. in terms of x and y. So also the solution is not necessarily an expression of either variable as a function of the other, but is generally a relation between x and y which makes either an implicit function of the other. When we recognize the left member of equation (3) as an " exact differential," that is, the differential of some function of x and y, the solution is obvious. For example, given the equa- tion xdy-\-ydx = o, (4) the solution xy = C, (5) where C is an arbitrary constant, is obtained by " direct inte- gration." When a particular value is attributed to (7, the result is a " particular integral ; " thus^ = x~ l is a particular integral of equation (4), while the more general relation expressed by equation (5) is known as the " complete integral." In general, the given expression Mdx -j- Ndy is not an ex- act differential, and it is necessary to find some less direct method of solution. The most obvious method of solving a differential equation of the first order and degree is, when practicable, to " separate the variables," so that the coefficient of dx shall contain x only, and that of dy, y only. For example, given the equation ( i y)dx -f- (i -{- x}dy = o, (6) the variables are separated by dividing by (\-\-x](\ y). dx dy Thus i h =0, I + x ' I y Each term is now directly integrable, and hence log (I +*) log (i y) = c. The solution here presents itself in a transcendental form, but it is readily reduced to an algebraic form. For, taking the exponential of each member, we find t ^ = c, whence i -J- x = C(i y) t (7) where C is put for the constant ^. GEOMETRICAL REPRESENTATION. 3 To verify the result in this form we notice that differentia- tion gives dx = Cdy, and substituting in equation (6) we find - C(i -y) + i +* = o, which is true by equation (7). Prob. i. Solve the equation dy -\- y tan x dx = o. Ans. y=C cos x. Prob. 2. Solve ^ + '/ = a\ Ans. . dx by a -, , ^ / + 1 Prob. 3. Solve -j- = =VT Ans. y = ex Prob. 4. Helrnholtz's equation for the strength of an electric current C at the time / is C = - - ~ R R dt* where , R, and L are given constants. Find the value of C, de- termining the constant of integration by the condition that its initial value shall be zero. ART. 2. GEOMETRICAL REPRESENTATION. The meani'ng of a differential equation may be graphically illustrated by supposing simultaneous values of x and y to be the rectangular coordinates of a variable point. It is conven- ient to put/ for the value of the ratio dy-.dx. Then P being the moving point (x, y) and denoting the inclination of its path to the axis of x, we have dy p = -j- = tan 0. dx The given differential equation of the first order is a relation between/, x, andy, and, being of the first degree with respect to/, determines in general a single value of/ for any assumed values of x and y. Suppose in the first place that, in addition to the differential equation, we were given one pair of simul- taneous' values of x and y, that is, one position of the point P. Now let P start from this fixed initial point and begin to move in either direction along the straight line whose inclination DIFFERENTIAL EQUATIONS. is determined by the value of p corresponding to the initial values of x and y. We thus have a moving point satisfying the given differential equation. As the point P moves the values of x and y vary, and we must suppose the direction of its motion to vary in such a way that the simultaneous values of x, y, and/ continue to satisfy the differential equation. In that case, the path of the moving point is said to satisfy the differential equation. The point P may return to its initial position, thus describing a closed curve, or it may pass to infin- ity in each direction from the initial point describing an infinite branch of a curve.* The ordinary cartesian equation of the path of P is a particular integral of the differential equation. If no pair of associated values of x and y be known, P may be assumed to start from any initial point, so that there is an unlimited number of curves representing particular integrals of the equation. These form a "system of curves," and the complete integral is the equation of the system in the usual form of a relation between x, y, and an arbitrary " parameter." This parameter is of course the constant of integration. It is constant for any one curve of the system, and different values of it determine different members of the system of curves, or different particular integrals. As an illustration, let us take equation (4) of Art. I, which may be written d y___y_ dx x' Denoting by the inclination to the axis of x of the line joining P with the origin, the equation is equivalent to tan = tan 6, and. therefore expresses that P moves in a direction inclined equally with OP to either axis, but on the other * When the form of the functions M and N is unrestricted, there is no reason why either of these cases should exist, but they commonly occur among such differential equations as admit of solution. PRIMITIVE OF A DIFFERENTIAL EQUATION. 5 side. Starting from any position in the plane, the point P thus moving must describe a branch of an hyperbola having the two axes as its asymptotes ; accordingly, the complete integral xy = C is the equation of the system consisting of these hyperbolas. ff^\ Lc- Prob. 5. Write the differential equation which requires P to move in a direction always perpendicular to OP, and thence derive the equation of the system of curves described. Ans ^-_*.*' + v'-C ';<** ,'* + * * Prob. 6. What is the system described when is the comple- * . ment of ? 6 - Z^ Ans. x 9 -/ = C. oJ^jv^TProb. 7. If "^OsJ^, show geometrically that the system described 5 ,^^consists of circles, and find the differential equation. ^ 6 4(|F Ans. 2xydx = (x* y*)dy. ART. 3. PRIMITIVE OF A DIFFERENTIAL EQUATION. Let us now suppose an ordinary relation between x and y, which may be represented by a curve, to be given. By differ- entiation we may obtain an equation of which the given equa- , tion is of course a solution or particular integral. But by combining this with the given equation any number of differ- ential equations of which the given equation is a solution may be found. For example, from y = m(x a) (l) we obtain directly 2ydy = mdx, (2) of which equation (i) is an integral; again, dividing (2) by (i) we have *** y x-a' and of this equation also (i) is an integral. If in equation (i) m be regarded as an arbitrary parameter, it is the equation of a system of parabolas having a common axis and vertex. The differential equation (3), which does not contain m, is satisfied by every member of this system of curves, DIFFERENTIAL EQUATIONS. Hence equation (i) thus regarded is the complete integral of equation (3), as will be found by solving the equation in which the variables are already separated. Now equation (3) is obviously the only differential equation independent of m which could be derived from (i) and (2), since it is the result of eliminating ;;/. It is therefore the " differ- ential equation of the system ; " and in this point of view the integral equation (i) is said to be its "primitive." Again, if in equation (i) a be regarded as the arbitrary con- stant, it is the equation of a system of equal parabolas having a common axis. Now equation (2) which does not contain a is satisfied by every member of this system of curves; hence it is the differential equation of the system, and its primitive is equation (i) with a regarded as the arbitrary constant. Thus, a primitive is an equation containing as well as x and" y an arbitrary constant, which we may denote by C, and the corresponding differential equation is a relation between x, y, and/, which is found by differentiation, and elimination of C if necessary. This is therefore also a method of verifying the com- plete integral of a given differential equation. For example, in verifying the complete integral (7) in Art. i we obtain by differ- entiation i = Cp. If we use this to eliminate C from equa- tion (7) the result is equation (6); whereas the process before employed was equivalent to eliminating / from equation (6), thereby reproducing equation (7). Prob. 8. Write the equation of the system of circles in Prob. 7, Art. 2, and derive the differential equation from it as a primitive. Prob. 9. Write the equation of the system of circles passing through the points (o, b} and (o, b], and derive from it the differ- ential equation of the system. ART. 4. EXACT DIFFERENTIAL EQUATIONS. In Art. I the case is mentioned in which Mdx -(- Ndy is an " exact differential," that is, the differential of a function of x andj. Let u denote this function; then (i) EXACT DIFFERENTIAL EQUATIONS. and in the notation of partial derivatives M=* t N=^. 'dx -dy 3* Then, since by a theorem of partial derivatives r- -dy -dx This condition rnjj^t therefore be fulfilled by M and N in order that equation (i) may be possible. When it is fulfilled Mdx -\- Ndy o is said to be an " exact differential equation," and its complete integral is u = C. (3) For example, given the equation x(x -\- 2y]dx -f- (x* y*}dy = o, = 2x, and - = 2*; the condition (2) is fulfilled, and the equation is exact. To find the function ?/, we may integrate Mdx, treating y as a constant; thus, &+Sy = Y, in which the constant of integration Y may be a function of y. The result of differentiating this is ;* - x*dx -\- 2xy dx -f- x*dy = dY, which should be identical with the given equation ; therefore, dY = y*dy, whence Y = $y a -}- C, and substituting, the com- plete integral may be written The result is more readily obtained if we notice that all | terms containing x and dx only, or y and dy only, are exact w differentials; hence it is only necessary to examine the termsj containing both x and y. In the present case, these are 2xy dx -\- x*dy, which obviously form the differential of x*y ; whence, integrating and multiplying by 3, we obtain the result above. The complete integral of any equation, in whatever way it '8 DIFFERENTIAL EQUATIONS. was found, can be put in the form u = C, by solving for C. Hence an exact differential equation du = o can be obtained, which must be equivalent to the given equation Mdx -f- Ndy o, (4) here supposed not to be exact. The exact equation du = o must therefore be of the form }ji(Mdx + Ndy) = o, (5) where /* is a factor containing at least one of the variables x and/. Such a factor is called an " integrating factor" of the given equation. For example,- the result of differentiating equation (7), Art. I, when put in the form u = C, is (i -y)dx + (\ 4- x)dy _ (i - yf so that (i y)~* is an integrating factor of equation (6). It is to be noticed that the factor by which we separated the variables, namely, (i y)~\i x)~ l , is also an integrating factor. It follows that if an integrating factor can be discovered, the given differential equation can at once be solved.* Such a factor is sometimes suggested by the form of the equation. Thus, given (y x)dy -\-ydx = o, the terms ydx xdy, which contain both x and y, are not ex- act, but become so when divided by either x 1 or y*\ and be- cause the remaining term contains^ only, j/~ 2 is an integrating factor of the whole expression. The resulting integral is ig.y + - = c. Prob. 10. Show from the integral equation in Prob. 9, Art. 3, that x~* is an integrating factor of the differential equation. Prob. n. Solve the equation x(x' 1 + 37*)^ -\-y(y* -f 3* a )^ = o. Ans. x* * Since f*M and uN in the exact equation (5) must satisfy the condition (2), we have a partial differential equation for //; but as a general method of finding ft this simply comes back to the solution of the original equation. HOMOGENEOUS EQUATION. , xdyydx Prob. 12. Solve the equation ydy -\-xdx -j -- . , . = o. * (Ans. 2 X Prob. 13. If w = c is a form of the complete integral and p the corresponding integrating factor, show that l*f(u) is the general expression for the integrating factors. Prob. 14. Show that the expression x a }P(mydx + nxdy) has the integrating factor jp**- 1 -*^**- 1 .-^; an d by means of such a factor solve the equation y(y* + 2x*)dx -(- jc(^ 4 2y)</v = o. Ans. 2x*y y* = ex*. V Prob. 15. Solve (x 1 -{- y'}dx 2xydy o. Ans. x* y* = ex. ART. 5. HOMOGENEOUS EQUATION. The differential equation Mdx -f- Ndy = o is said to be homogeneous when M and N are homogeneous functions of v and y of the same degree ; or, what is the same thing, when dy y Hr is expressible as a function of . If in such an equation the variables are changed from x and y to x and v, where y v = whence y = xv and dy = xdv -f- vdx, the variables x and # will be separable. For example, the equation (x 2y)dx -f- ydy = o is homogeneous; making the substitutions indicated and dividing by x, (i 2v)dx -f- v(xdv -|- ^^f) = O, dx vdv whence -- \- -, -. = o. x ' (y i/ Integrating, log x + log (v - i) = C\ and restoring^, x log ( _y *) -- - = C. ' y x The equation Mdx -f- -/VWy = o can always be solved when 10 DIFFERENTIAL EQUATIONS. M and N are functions of the first degree, that is, when it is of the form (ax + fy + c]dx + (a'x -f b'y + c'}dy = o. For, assuming x = x' -f- //, y = y' -f- ^> it becomes (*'+ b'y'+ ah + bk+ c)dx'+(a'x'+ b'y'+ a'h +b'k+c')dy' =o, which, by properly determining h and k, becomes (ax' + /X*' + (a'x' -f- '/>//, a homogeneous equation. This method fails when a : b == a' : b', that is, when the aquation takes the form (ax -\-by-\- c)dx -\- \m(ax -f- by) -\- c'~\dy = o ; but in this case if we put z = ax -f- by, and eliminate y, it will be found that the variables x and 2 can be separated. Prob. 16. Show that a homogeneous differential equation repre- sents a system of similar and similarly situated curves, the origin being the center of similitude, and hence that the complete integral may be written in a form homogeneous in x, y, and c. Prob. 17. Solve xdy ydx y(x* -f y*)dx = o. Ans. x* = * 2cy. Prob. 1 8. Solve (3^ 7* + i)dx + (jy 3* + $)dy = o. Ans. (y x + r)'(7 + x i) 5 = r. Prob. 19. Solve (x*.-{-y*)dx zxydy = o. Ans. .r a y 9 = ex. * Prob. 20. Solve (i + xy)ydx + (i tfy).*^ = o by introducing _ the new variable z = xy. Ans. x = Cye*y. Prob. 21. Solve ~ ax -\-by-\-c. Ans. ata+^-i- -!-& = <?**. ART. 6. THE LINEAR EQUATION. A differential equation is said to be "linear" when (one of the variables, say x, being regarded as independent,) it is of the first degree with respect to y, and its derivatives. The linear equation of the first order may therefore be written in the form THE LINEAR EQUATION. 11 where P and Q are functions of x only. Since the second member is a function of x, an integrating factor of the first member will be an integrating factor of the equation provided it contains x only. To find such a factor, we solve the equation dy which is done by separating the variables ; thus, = Pdx \ whence log y = c I Pdx or y = Ce ~' M "- (3) Putting this equation in the form u = c, the corresponding exact equation is e SPd \dy + Pydx} = o, whence tr ' ' is the integrating factor required. Using this factor, the general solution of equation (i) is Qdx + C. (4) In a given example the integrating factor should of course be simplified in form if possible. Thus (i -(- x*}dy = (m -\- xy)dx is a linear equation for/; reduced to the form (i), it is dy x m dx i -\-a? i -f- x" from which The integrating factor is, therefore, e fpdx = whence the exact equation is dy xy dx mdx r J I 12 DIFFERENTIAL EQUATIONS. Integrating, there is found y w* ~ v(i+O =: i/(i + *') + C) or y = mx -\- C \/(l + X*). ri r An equation is sometimes obviously linear, not for^, but for some function of y.' For example, the equation dy . -j \- tan y = x sec y when multiplied by cos y takes a form linear for sin y ; the integrating factor is e*, and the complete integral sin y = x i -|- ce~ x . dy In particular, the equation ^- -j- Py = Qy* t which is known as " the extension of the linear equation," is readily put in a form linear for y l ~ M . d y , . rob. 22. Solve ar-r- + (i 2#)y = ^ a - Ans. y = x*(\ + ce*). dy Prob. 23. Solve cos x - -{-y i -(- sin ^r = o. Ans. Xsec * + tan *) = * + ^ A Prob. 24. Solve cos x + y sin ^ = i. rf^ Ans. y = sin * + c cos ^c. Prob. 25. Solve -^ = *V xy. Ans. 4 = ** + i +^. Prob. 26. Solve ^ = - ,' , 3 . Ans. - = 2 -/+ dx yy + 0:7 a: . y ART. 7. FIRST ORDER AND SECOND DEGREE. If the given differential equation of the first order, or re- lation between x, y, and /, is a quadratic for p, the first step in the solution is usually to solve for /. The resulting value of / will generally involve an irrational function of x and y\ >so that an equation expressing such a value of /, like some of those solved in the preceding pages, is not properly to be re- FIRST ORDER AND SECOND DEGREE. 13 garded as an equation of the first degree. In the exceptional case when the expression whose root is to be extracted is a perfect square, the equation is decomposable into two equa- tions properly of the first degree. For example, the equation y x when solved for / gives 2p = -, or 2p = ; it may therefore be written in the form (2px - y)(2py - x) = Q, and is satisfied by putting either dy _ y dy _ x ~T~ - OF ~7~ - - ax 2x ax 2y The integrals of these equations are y* = ex and 2y* x 1 = C, and these form two entirely distinct solutions of the given equation. As an illustration of the general case, let us take the equation Separating the variables and integrating, Vic Vy = Vc, (2) and this equation rationalized become^ (x- y y-2c(x+y)+c*=o. (3) There is thus a single complete integral containing one arbi- trary constant and representing a single system of curves; namely, in this case, a system of parabolas touching each axis at the same distance c from the origin. The separate equa- tions given in the form (2) are merely branches of the same parabola. Recurring now to the geometrical interpretation of a differ- ential equation, as given in Art. 2, it was stated that an equa- tion of the first degree determines, in general, for assumed values of x and y, that is, at a selected point in the plane, a single value of p. The equation was, of course, then supposed 14 DIFFERENTIAL EQUATIONS. rational in x and y* The only exceptions occur at points for which the value of p takes the indeterminate form ; that is, the equation being Mdx -\- Ndy = o, at points (if any exist) for which M = o and N = o. It follows that, except at such points, no two curves of the system representing the complete integral intersect, while through such points an unlimited num- ber of the curves may pass, forming a "pencil of curves." f On the other hand, in the case of an equation of the second cdegree, there will in general be two values of / for any given point. Thus from equation (i) above we find for the point (4, i), p = ; there are therefore two directions in which a point starting from the position (4, i) may move while satis- fying the differential equation. The curves thus described represent two of the particular integrals. If the same values of x and y be substituted in the complete integral (3), the re- sult is a quadratic for c, giving c = 9 and c = i, and these determine the two particular integral curves, Vx -j- Vy = 3, and Vx Vy = i. In like manner the general equation of the second degree, which may be written in the form where L, M, and A 7 " are one-valued functions of x and y, repre- sents a system of curves of which two intersect in any given point for which p is found to have two real values. For these points, therefore, the complete integral should generally give two real values of c. Accordingly we may assume, as the standard form of its equation, PC* * In fact / was supposed to be a one-valued function of x and y; thus, / = sin" 1 * would not in this connection be regarded as an equation of the first degree. f In Prob. 6, Art. 3, the integral equation represents the pencil of circles pass- ing through the points (o, b) and (o, b)\ accordingly/ in the differential equa- tion is indeterminate at these points. In some cases, however, such a point is merely a node of one particular integral. Thus in the illustration given in Art. 2, / is indeterminate at the origin, and this point is a node of the only particular integral, xy = o, which passes through it. SINGULAR SOLUTIONS. 15 where P, Q, and R are also one-valued functions of x and y. If there are points which make / imaginary in the differential equation, they will also make c imaginary in the integral. Prob. 27. Solve the equation /" +y = i and reduce the inte- ./ gral to the standard form. Ans. (y -f- cos x)^ 2C sin x -f- y cos x = o. Prob. 28. Solve yp* + zxp y = o, and show that the intersect- ing curves at any given point cut at right angles. , Prob. 29. Solve (x 9 + i)/ = i. Ans. <rV r - text? = i. ART. 8. SINGULAR SOLUTIONS. A differential equation not of the first degree sometimes admits of what is called a " singular solution ; " that is to say, a solution which is not included in the complete integral. For suppose that the system of curves representing the complete integral has an envelope. Every point A of this envelope is a point of contact with a particular curve of the complete in- tegral system ; therefore a point moving in the envelope when passing through A has the same values of x, y, and /as if it were moving through A in the particular integral curve. Hence such a point satisfies the differential equation and will continue to satisfy it as long as it moves in the envelope. The equation of the envelope is therefore a solution of the equation. As an illustration, let us take the system of straight lines whose equation is i a y = cx + -, (I) C where c is the arbitrary parameter. The differential equation derived from this primitive is y=px + - p * (2) of which therefore (i) is the complete integral. Now the lines represented by equation (i), for different values of c, are the tangents to the parabola / = V**- (3) 1G DIFFERENTIAL EQUATIONS. A point moving in this parabola has the same value of/ as if it were moving in one of the tan- gents, and accordingly equation (3) will be found to satisfy the differential equation (2). It will be noticed that for any point on the convex side of the parabola there are two real values of p ; for a point on the other side the values of / are imaginary, and for a point on the curve they are equal. Thus its equation (3) expresses the relation between x and y which must exist in order that (2) regarded as a quadratic for p may have'equal roots, as will be seen on solving that equation. In general, writing the differential equation in the form the condition of equal roots is o. (5) The first member of this equation, which is the " discrimi- nant " of equation (4), frequently admits of separation into factors rational in x and_y. Hence, if there be a singular solu- tion, its equation will be found by putting the discriminant of the differential equation, or one of its factors, equal to zero. It does not follow that every such equation represents a solu- tion of the differential equation. It can only be inferred that it is a locus of points for which the two values of / become equal. Now suppose that two distinct particular integral curves touch each other. At the point "of contact, the two values of/, usually distinct, become equal. The locus of such points is called a "tac-locus." Its equation plainly satisfies the discriminant, but does not satisfy the differential equation. An illustration is afforded by the equation SINGULAR SOLUTIONS. 17 of which the complete integral isy 1 -f- (x c)* = a*, and the discriminant, see equation (5), is j"(y tf 2 ) = o. This is satisfied by y = a, y a, and y = o, the first two of which satisfy the differential equation, while U = o does not. The complete integral represents in this case all circles of radius a with center on the axis of x. Two of these circles touch at every point of the axis of x, which is thus a tac-locus, while y = a and y = a constitute the envelope. The discriminant is the quantity which appears under the radical sign when the general equation (4) is solved for/, and therefore it changes sign as we cross the envelope. But the values of / remain .real as we cross the tac-locus, so that the discriminant cannot change sign. Accordingly the factor which indicates a tac-locus appears with an even exponent (as y 1 in the example above), whereas the factor indicating the singular solution appears as a simple factor, or with an odd exponent. A simple factor of the discriminant, or one with an odd ex- ponent, gives in fact always the boundary between a region of the plane in which / is real and one in which p is imaginary ; nevertheless it may not give a singular solution. For the two arcs of particular integral curves which intersect in a point on the real side of the boundary may, as the point is brought up to the boundary, become tangent to each other, but not to the boundary curve. In that case, since they cannot cross the boundary, they become branches of the same particular inte- gral forming a cusp. A boundary curve of this character is called a "cusp-locus" ; the value of / for a point moving in it is of course different from the equal values of/ at the cusp, and therefore its equation does not satisfy the differential equation.* Prob. 30. To what curve is the line y = mx -\- a |/(i m*) always tangent ? Ans. y 1 x* = a*. Prob. 31. Show that the discriminant of a decomposable differ- * Since there is no reason why the values of/ referred to should be identical, we conclude that the equation Z/ 9 -f Mp + N = o has not in general a singular solution, its discriminant representing a.cusp-locus except when a certain con- dition is fulfilled. 18 DIFFERENTIAL EQUATIONS. cntial equation cannot be negative. Interpret the result of equating it to zero in the illustrative example at the beginning of Art. 7. Prob. 32. Show that the singular solutions of a homogeneous dif- ferential equation represent straight lines passing through the origin. Prob. 33. Solve the equation xp* zyp -\- ax o. Ans. x* 2cy -f- af = o ; singular solution y = ax*. Prob. 34. Show that the equation /" -f- zxp y = o has no sin- gular solution, but has a cusp-locus, and that the tangent at every cusp passes through the origin. ART. 9. SINGULAR SOLUTION FROM COMPLETE INTEGRAL. When the complete integral of a differential equation of the second degree has been found in the standard form PS+Qc + R = o (i) (see the end of Art. 7), the substitution of special values of x and y in the functions P, Q, and R gives a quadratic for c whose roots determine the two particular curves of the system which pass through a given point. If there is a singular solution, that is, if the system of curves has an envelope, the two curves which usually intersect become identical when the given point is moved up to the envelope. Every point on the en- velope therefore satisfies the condition of equal roots for equa- tion (i), which is Q- 4/^ = 0; (2) and, reasoning exactly as in Art. 8, we infer that the equation of the singular solution will be found by equating to zero the discriminant of the equation in c or one of its factors. Thus the discriminant of equation (i), Art. 8, or "^-discriminant," is the same as the "/-discriminant," namely, y ^ax, which equated to zero is the equation of the envelope of the system of straight lines. But, as in the case of the /-discriminant, it must not be inferred that every factor gives a singular solution. For ex- ample, suppose a squared factor appears in the ^-discriminant. The locus on which this factor vanishes is not a curve in cross- ing which c and/ become imaginary. At any point of it there SINGULAR SOLUTION FROM COMPLETE INTEGRAL. 19 \vill be two distinct values of p, corresponding to arcs of par- ticular integral curves passing through that point ; but, since there is but one value of c, these arcs belong to the same par- ticular integral, hence the point is a double point or node. The locus is therefore called a " node-locus." The factor repre- senting it does not appear in the /-discriminant, just as that representing a tac-locus does not appear in the <r-discriminant. Again, at any point of a cusp-locus, as shown at the end of Art. 8, the two branches of particular integrals become arcs ot the same particular integral ; the values of c become equal, so that a cusp-locus also makes the ^-discriminant vanish. The conclusions established above obviously apply also to equations of a degree higher than the second. In the case of the <r-equation the general method of obtaining the condition for equal roots, which is to eliminate c between the original and the derived equation, is the same as the process of finding the envelope or "locus of the ultimate intersections" of a, system of curves in which c is the arbitrary parameter. Now suppose the system of curves to have for all values of * a double point, it is obvious that among the intersections of two neighboring curves there are two in the neighborhood of the nodes, and that ultimately they coincide with the node, which accounts for the node-locus appearing twice in the dis- criminant or locus of ultimate intersections. In like manner, * It is noticed in the second foot-note to Art. 7 that for an equation of the first degree p takes the indeterminate form, not only at a point through which all curves of the system pass (where the value of c would also be found indeter- minate), but at a node of a particular integral. So also when the equation is of the th degree, if there is a node for a particular value of c, the n values of e at the point (which is not on a node-locus where two values of c are equal) deter- mine n -f- i arcs of particular integrals passing through the point ; and there- fore there are n -\- i distinct values of / at the point, which can only happen when p takes the indeterminate form, that is to say, when all the coefficients of the/-eguation (which is of the wth decree} vanish. See Cayley on Singular So- lutions' in the Messenger of Mathematics. New Series, Vol. II, p. 10 (Collected Mathematical Works, Vol. VIII, p. 529). The present t-neory of Singular Solu- tions was established by Cayley in this paper and its continuation, Vol. VI, p. 23. See also a paper by Dr. Glaisher, Vol. XII, p. I. 20 DIFFERENTIAL EQUATIONS. if there is a cusp for all values of c, there are three intersections of neighboring curves (all of which may be real) which ulti- mately coincide with the cusp ; therefore a cusp-locus will appear as a cubed factor in the discriminant.* Prob. 35. Show that the singular solutions of a homogeneous equation must be straight lines passing through the origin. Prob. 36. Solve 3/V 2xyp + 47* x* = o, and show that there is a singular solution and a tac-focus. Prob. 37. Solve yp*-\- 2Xp y = o t and show that there is an imaginary singular solution. Ans. y* = zcx + c 1 . Prob. 38. Show that the equation (i x^p' = i y* represents a system of conies touching the four sides of a square. Prob. 39. Solve yp* ^xp -\-y = o ; examine and interpret both discriminants. Ans. c 1 + 2cx($y' 8.x 2 ) 3* a y +/* = o. ART. 10. SOLUTION BY DIFFERENTIATION. The result of differentiating a given differential equation o the first order is an equation of the second order, that is, it d*y contains the derivative -r-^ ; but, if it does not contain y ex- plicitly, it may be regarded as an equation of the first order for the variables x and/. If the integral of such an equation can be obtained it will be a relation between x, p, and a constant of integration c, by means of which / can be eliminated from the original equation, thus giving the relation between x, y> and c which constitutes the complete integral. For example, the equation (i) * The discriminant of PC* -\- Qc + R = o represents in general an envelope, no further condition requiring to be fulfilled as in the case of the discriminant of Z/ 2 -f- Mp -f- -A/ = o. Compare the foot-note to Art. 8. Therefore where there is an integral of this form there is generally a singular solution, although Z/ 9 -f- Mp -}- .A 7 = o has not in general a singular solution. We conclude, there- fore, that this equation (in which Z, M, and N are one-valued functions of x and y) has not in general an integral of the above form in which P, Q, and R are one-valued functions of x and y. Cayley, Messenger of Mathematics, New Series, Vol. VI, p. 23. SOLUTION BY DIFFERENTIATION. 21 when solved for^, becomes y = x + ^p\ (2) whence by differentiation The variables can be separated in this equation, and its inte- gral is Substituting in equation (2), we find which is the complete integral of equation (i). This method sometimes succeeds with equations of a higher degree when the solution with respect to p is impossible or leads to a form which cannot be integrated. A differential equation between p and one of the two variables will be ob- tained by direct integration when only one of the variables is explicitly present in the equation, and also when the equation is of the first degree with respect to x and y. In the latter case after dividing by the coefficient of y, the result of differ- entiation will be a linear equation for x as a function of p, so that an expression for x in terms of p can be found, and then by substitution in the given equation an expression for y in terms of p. Hence, in this case, any number of simultaneous values of x and y can be found, although the elimination of p may be impracticable. In particular, a homogeneous equation which cannot be solved for p may be soluble for the ratio y : x, so as to assume the form y = x(f>(p). The result of differentiation is in which the variables x and p can be separated. Another special case is of the form y = P* +/<, (0 22 DIFFERENTIAL EQUATIONS. which is known as Clairaut's equation. The result of differ- entiation is which implies either =o, or The elimination of p from equation (i) by means of the first of these equations * gives a solution containing no arbi- trary constant, that is, a singular solution. The second is a differential equation for/; its integral is p = c, which in equation (i) gives the complete integral y = c X +f(c\ (2} This complete integral represents a system of straight lines, the singular solution representing the curve to which they are all tangent. An example has already been given in Art. 8. A differential equation is sometimes reducible to Clairaut's form by means of a more or less obvious transformation of the variables. It may be noticed in particular that an equation of the form y = nx p _}_ (j)( x , p) is sometimes so reducible by transformation to the independent variable z, where x = z n ; and an equation of the form by transformation to the new dependent variable v y n . A double transformation of the form indicated may succeed when the last term is a function of both x and y as well as of/. Prob. 40. Solve the equation $y = 2/ 3 + 3/ 2 ; find a singular solution and a cusp-locus. Ans. (x -j- y -\- c i) a == ( xJ t~ f Y' a Prob. 41. Solve zy = xp -\ , and find a cusp-locus. Ans. aV \2acxy -f- Scy 3 i2x*y* -\- i6ax* == o. * The equation is in fact the same that arises in the general method for the condition of equal roots. See Art. 9. GEOMETRIC APPLICATIONS ; TRAJECTORIES. 23 Prob. 42. Solve (x* (?}p* 2xyp + / a* = o. Ans. The circle x* + / = a\ and its tangents. Prob. 43. Solve y = xp + *y. Ans. ^ -j- <r .ay = o, and i + $x*y = o. Prob. 44. Solve / s $xyp + By 1 = o. Ans. y = c(x c}* 2jy = 43? and y = o are singulat solutions \y = o is also a particular integral. Prob. 45. Solve x\y px] = yp\ Ans. / = ex* + c\ ART. 11. GEOMETRIC APPLICATIONS ; TRAJECTORIES. Every property of a curve which involves the direction of its tangents admits of statement in the form of a differential equation. The solution of such an equation therefore deter- mines the curve having the given property. Thus, let it be required to determine the curve in which the angle between the radius vector and the tangent is n times* the vectorial angle. Using the expression for the trigonometric tangent of that angle, the expression of the property in polar coordi- nates is rdB = tan nO. dr Separating the variables and integrating, the complete integral is r" = C H sin nB. The mode in which the constant of integration enters here shows that the property in question is shared by all the mem- bers of a system of similar curves. The solution of a question of this nature will thus in gen- eral be a system of curves, the complete integral of a differential equation, but it may be a singular solution. Thus, if we ex- press the property that the sum of the intercepts on the axes made by the tangent to a curve is equal to the constant a, the straight lines making such intercepts will themselves consti- tute the complete integral system, and the curve required is the singular solution, which, in accordance with Art. 8, is the 24 DIFFERENTIAL EQUATIONS. envelope of these lines. The result in this case will be found to be the parabola Vx ~\- \^y = Va. An important application is the determination of the "orthogonal trajectories" of a given system of curves, that is to say, the curves which cut at right angles every\curve of the given system. The differential equation of the trajectory is readily derived from that of the given system ; for at every point of the trajectory the value of p is the negative reciprocal of its value in the given differential equation. We have there- fore only to substitute /"' for p to obtain the differential equation of the trajectory. For example, let it be required to determine the orthogonal trajectories of the system of pa- rabolas having a common axis and vertex. The differential equation of the system found by eliminating a is 2 xdy = y dx. Putting -- in place of -7-, the differential equation oi dy dx the system of trajectories is 2.x dx -\- ydy = o, Whence, integrating, , The trajectories are therefore a system of similar ellipses with axes coinciding with the coordinate axes. Prob. 46. Show that when the differential equation of a system is of the second degree, its discriminant and that of its trajectory system will be identical ; but if it represents a singular solution in one system, it will constitute a cusp locus of the other. Prob. 47. Determine the curve whose subtangent is constant and equal to a. Ans. ce*=y*. Prob. 48. Show that the orthogonal trajectories of the curves r n =c" sin## are the same system turned through the angle about 2n the pole. Examine the cases n = i, n = 2, and n = . Prob. 49. Show that the orthogonal trajectories of a system of SIMULTANEOUS DIFFERENTIAL EQUATIONS. 25 circles passing through two given points is another system of circles having a common radical axis. Prob. 50. Determine the curve such that the area inclosed by any two ordinates, the curve and the axis of x, is equal to the product of the arc and the constant line a. Interpret the singular solution. Ans. The catenary ^ = \a(e a e a ). Prob. 51. Show that a system of confocal conies is self-orthog- onal. ART. 12. SIMULTANEOUS DIFFERENTIAL EQUATIONS. A system of equations between n -j- I variables and their differentials is a " determinate" differential system, because it serves to determine the n ratios of the differentials ; so that, taking any one of the variables as independent, the others vary in a determinate manner, and may be regarded as functions of the single independent variable. Denoting the variables by .#, y, 2, etc., the system may be written in the symmetrical form dx _ dy _ dz _ ~X~~^Y~~Z~- "' where X, Y, Z . . . may be any functions of the variables. If any one of the several equations involving two differen- tials contains only the two corresponding variables, it is an ordinary differential equation ; and its integral, giving a re- lation between these two variables, may enable us by elimina- tion to obtain another equation containing two variables only, and so on until n integral equations have been obtained. Given, for example, the system dx_d^ _dz^ ,. x~ z " y' The relation between dy and dz above contains the varia- bles y and z only, and its integral is 7' s? = a. (2) Employing this to eliminate z from the relation between dx and dy it becomes dx _ dy ~' 20 DIFFERENTIAL EQUATIONS. of which the integral is *) = *>*' (3) The integral equations (2) and (3), involving two constants of integration, constitute the complete solution. It is in like manner obvious that the complete solution of a system of n equations should contain n arbitrary constants. Confining ourselves now to the case of three variables, an extension of the geometrical interpretation given in Art. 2 presents itself. Let x, y, and z be rectangular coordinates of P referred to three planes. Then, if P starts from any given position A, the given system of equations, determining the ratios dx : dy : dz, determines the direction in space in which P moves. As P moves, the ratios of the differentials (as deter- mined by the given equations) will vary, and if we suppose P to move in such a way as to continue to satisfy the differential equations, it will describe in general a curve of double curva- ture which will represent a particular solution. The complete solution is represented by the system of lines which may be thus obtained by varying the position of the initial point A. This system is a " doubly infinite " one ; for the two relations between x, y, and z which define it analytically must contain two arbitrary parameters, by properly determining which we can make the line pass through any assumed initial point.* Each of the relations between x, y and z, or integral equa- tions, represents by itself a surface, the intersection of the two surfaces being a particular line of the doubly infinite system. An equation like (2) in the example above, which contains only one of the constants of integration, is called an integral of the differential system, in contradistinction to an " integral equa- * It is assumed in the explanation that X, V, and Zare one-valued functions of x, y, and 2. There is then but one direction in which P can move when passing a given point, and the system is a non-intersecting system of lines. But if this is not the case, as for example when one of the equations giving the ratio of the differentials is of higher degree the lines may form an intersecting sys- tem, and there would be a theory of singular solutions, into which we do not here enter. SIMULTANEOUS DIFFERENTIAL EQUATIONS. 27 tion " like (3), which contains both constants. An integral represents a surface which contains a singly infinite system of lines representing particular solutions selected from the doubly infinite system. Thus equation (2} above gives a surface on which lie all those lines for which a has a given value, while b may have any value whatever ; in other words, a surface which passes through an infinite number of the particular solution lines. The integral of the system which corresponds to the con- stant b might be found by eliminating a between equations (2) and (3). It might also be derived directly from equation (i) ; thus we may write dx^ _ dy _dz _ dy -J- dz _du x z y y ' -f- z u' in which a new variable u =. y -f- z is introduced. The rela- tion between dx and du now contains but two variables, and its integral, y + z = bx, (4) is the required integral of the system ; and this, together with the integral (2\ presents the solution of equations (i) in its standard form. The form of the two integrals shows that in this case the doubly infinite system of lines consists of hyper- bolas, namely, the sections of the system of hyperbolic cylinders represented by (2) made by the system of planes represented by (4). A system of equations of which the members possess a cer- tain symmetry may sometimes be solved in the following manner. Since dx _ dy _ dz _ \dx -f- pdy -f- vdz ' if we take multipliers A, /v, v such that we shall have \dx -f- pdy -f- y dz = o. If the expression in the first member is an exact differential, 28 DIFFERENTIAL EQUATIONS. direct integration gives an integral of the given system. For example, let the given equations be dx dy dz mz ny nx Iz ly mx ' /, m and n form such a set of multipliers, and so also do x, y and z. Hence we have Idx -j- mdy -f- ndz = o, and also xdx -\-ydy-\- zdz = o. Each of these is ai\ exact equation, and their integrals Ix -J- my -\- nz = a a nd x* -J- y* -j- z* = b* constitute the complete solution. The doubly infinite system of lines consists in this case of circles which have a common axis, namely, the line passing through the origin and whose direction cosines are proportional to /, m, and n. dx dy dz Prob. 132. Solve the equations -5 5 . = ^ = , and x y z 2xy 2xz interpret the result geometrically. (Ans. y=az, x*-\-y* -^-z^bz.} dx dy dz ^Prob. 53. Solve x z Prob. 54. Solve ,, ~ = , ^ = , ^- . (b c}yz (c a)zx (a b)xy Ans. x* +/ + 2 a = A, ax' -f b? + c# - B. ART. 13. EQUATIONS OF THE SECOND ORDER. A relation between two variables and the successive deriva- tives of one of them with respect to the other as independent variable is called a differential equation of the order indicated by the highest derivative that occurs. For example, is an equation of the second order, in which x is the independent EQUATIONS OF THE SECOND ORDER. 29 variable. Denoting as heretofore the first derivative by/, this equation may be written / , v\^P_\ j. i _ / \ and this, in connection with which defines /, forms a pair of equations of the first order, connecting the variables x, y, and /. Thus any equation of the second order is equivalent to a pair of simultaneous equations of the first order. When, as in this example, the given equation "does not con- tain^ explicitly, the first of the pair of equations involves only the two variables x and/ ; and it is further to be noticed that, when the derivatives occur only in the first degree, it is a linear equation for/. Integrating equation (i) as such, we find and then using this value of/ in equation (2), its integral is y = c, - mx + c t log \x + y(i + *')], (4) in which, as in every case of two simultaneous equations of the first order, we have introduced two constants of integration. An equation of the first order is readily obtained also when the independent variable is not explicitly contained in the equation. The general equation of rectilinear motion in ffs dynamics affords an illustration. This equation is = f(s), where s denotes the distance measured from a fixed center of d*v force upon the line of motion. It may be written = f(s), in U-f connection with = v, which defines the velocity. Eliminat- dt ing dt from these equations, we have vdv = f(s)ds, whose integral is $v* = I f(s)ds -\- c, the "equation of energy" for the unit mass. The substitution of the value found for v in the 30 DIFFERENTIAL EQUATIONS. second equation gives an equation from which t is found in terms of s by direct integration. The result of the first integration, such as equation (3) above, is called a "first integral" of the given equation of the second order ; it contains one constant of integration, and its complete integral, which contains a second constant, is also the "com- plete integral " of the given equation. A differential equation of the second order is " exact " when, all its terms being transposed to the first member, that member is the derivative with respect to x of an expression of the first order, that is, a function of x, y and p. It is obvious that the terms containing the second derivative, in such an exact differ- ential, arise solely from the differentiation of the terms con- taining/ in the function of x, y and/. For example, let it be required to ascertain whether is an exact equation. The terms in question are (i x*}-f-, ax which can arise only -from the differentiation of (i x^p. Now subtract from the given expression the complete deriva- tive of (i x*}p, which is , cTy dy /T _ y- \ - 2 X * \ / J a _/ ax ax the remainder is x -\- y, which is an exact derivative, namely, ax that of xy. Hence the given expression is an exact differ- ential, and (i-^ + ^y^ (6) is the first integral of the given equation. Solving thi-s linear -equation for y, we find the complete integral y = Cl x + c t tf(i *?). (7) ,. \Prob. 55- Solve (i - * 2 ) Ans. y = (sin* 1 #)" + c, sin" 1 x + c v THE TWO FIRST INTEGRALS. 31 * Prob. 56. Solve = . Ans. y = - + cjc\ dx x x v Prob. 57. Solve -^ = cfx tfy. Ans. c?x b*y A sin for -f- .5 cos for. vprob. 58. Solve y + ' = i. Ans. / = ** + ^ + c v ART. 14. THE Two FIRST INTEGRALS. We have seen in the preceding article that the complete integral of an equation of the second order is a relation be- tween x, y and two constants c t and c 9 . Conversely, any rela- tion between x, y and two arbitrary constants may be regarded as a primitive, from which a differential equation free from both arbitrary constants can be obtained. The process consists in first. obtaining, as in Art. 3, a differential equation of the first order independent of one of the constants, say c 9 , that is, a rela- tion between x, y,p and <:, , and then in like manner eliminating r, from the derivative of this equation. The result is the equa- tion of the second order or relation between x, y, p and q (q denoting the second derivative), of which the original equation is the complete primitive, the equation of the first order being the first integral in which c l is the constant of integration. It is obvious that we can, in like manner, obtain from the primi- tive a relation between x, y, p and c 9 , which will also be a first integral of the differential equation. Thus, to a given form of the primitive or complete integral there corresponds two first integrals. Geometrically the complete integral represents a doubly infinite system of curves, obtained by varying the values of c t and of independently. If we regard c t as fixed and c t as arbitrary, we select ffom that system a certain singly infinite system ; the first integral containing c, is the differential equa- tion of this system, which, as explained in Art. 2, is a relation between the coordinates of a moving point and the direction of its motion common to all the curves of the system. But DIFFERENTIAL EQUATIONS. the equation of the second order expresses a property involv- ing curvature as well as direction of path, and this property being independent of c l is common to all the systems corre- sponding to different values of c lt that is, to the entire doubly infinite system. A moving point, satisfying this equation, may have any position and move in any direction, provided its path has the proper curvature as determined by the value of q derived from the equation, when the selected values of x, y and/ have been substituted therein.* For example, equation (7) of the preceding article repre- sents an ellipse having its center at the origin and touching the lines x = I, as in the diagram ; c 1 is the ordinate of the point -of contact with x = i, and c 9 that of the point in which the ellipse cuts the axis of y. If we regard ^, as fixed and c, as arbitrary, the equation represents the system of ellipses touching the two lines at fixed points, and equation (6) is the differential equation of this system. In like manner, if , is fixed and c 1 arbitrary, equation (7) represents a system of ellipses cutting the axis of y in fixed points and touching the lines x= i. The corresponding differential equation will be found to be Finally, the equation of the second order, independent of , and [(5) of the preceding article] is the equation of the doubly infinite system of conies f with center at the origin, and touching the fixed lines x i,. * If the equation is of the second or higher degree in q, the condition for equal roots is a relation between x, y and/, which may be found to satisfy the given equation. If it does, it represents a system of singular solutions; each of the curves of this system, at each of its points, not only touches but osculates with a particular integral curve. It is to be remembered that a singular solu- tion of a first integral is not generally a solution of the given differential equa- tion; for it represents a curve which simply touches but does not osculate a set of curves belonging to the doubly infinite system. f Including hyperbolas corresponding to imaginary values of c*. THE TWO FIRST INTEGRALS. 33 But, starting from the differential equation of second order, we may find other first integrals than those above which corre- spond to , and a . For instance, if equation (5) be multiplied by/, it becomes which is also an exact equation, giving the first integral in which c t is a new constant of integration. Whenever two first integrals have thus been found inde- pendently, the elimination of / between them gives the com- plete integral without further integration.* Thus the result of eliminating p between this last equation and the first inte- gral containing c l [equation (6), Art. 13] is / -2c,xy + cfx 1 = C? - c t \ which is therefore another form of the complete integral. It is obvious from the first integral above that c t is the maximum value of y, so that it is the differential equation of the system of ellipse inscribed in the rectangle drawn in the diagram. A comparison of the two forms of the complete integral shows that the relation between the constants is c* = c? -j- c*. If a first integral be solved for the constant, that is, put in the form <j>(x, y, p) = c, the constant will disappear on differ- entiation, and the result will be the given equation of second order multiplied, in general, by an integrating factor. We can thus find any number of integrating factors of an equation already solved, and these may suggest the integrating factors of more general equations, as illustrated in Prob. 59 below. * The principle of this method has already been applied in Art. 10 to the solution of certain equations of the first order; the process consisted of forming the equation of the second order of which the given equation is a first integral (but with a particular value of the constant), then finding another first integral and deriving the complete integral by elimination of /. 31 DIFFERENTIAL EQUATIONS. Prob. 59. Solve the equation y + c?y = o in the form y = A cos ax + B sin 0.x; and show that the corresponding integrating factors are also inte- grating factors of the equation where X is any function of x; and thence derive the integral of this equation. /* /* Ans. <y sin ax I cos ax . Xdx cos ax I sin fl.r . Xdx. Prob. 60. Find the rectangular and also the polar differential ^equation of all circles passing through the origin. ART. 15. LINEAR EQUATIONS. A linear differential equation of any order is an equation of the first degree with respect to the dependent variable y and each of its derivatives, that is, an equation of the form where the coefficients />,... P n and the second member X are functions of the independent variable only. The solution of a linear equation is always supposed to be in the f orm y =f(x)\ and if j, is a function which satisfies the equation, it is customary to speak of the function j,, rather than of the equation y = jj/,, as an "integral" of the linear equa- tion. The general solution of the linear equation of the first order has been given in Art. 6. For orders higher than the first the general expression for the integrals cannot be effected by means of the ordinary functional symbols and the integral sign, as was done for the first order in Art. 6. The solution of equation (i) depends upon that of LINEAR EQUATIONS. 36 The complete integral of this equation will contain arbi- trary constants, and the mode in which these enter the expres- sion for y is readily inferred from the form of the equation. For let y l be an integral, and c t an arbitrary constant ; the re- sult of putting y = cjf 1 in equation (2) is ^, times the result of putting y = y l ; that is, it is zero ; therefore c l y l is an integral. So too, if y t is an integral, j/ a is an integral ; and obviously also c l y l -\- c^y t is an integral. Thus, if n distinct integrals/,, y t ,. . . y n can be found, y = Wi + w* + - + c y (3) will satisfy the equation, and, containing, as it does, the proper number of constants, will be the complete integral. Consider now equation (i); let Fbe a particular integral of it, and denote by u the second member of equation (3), which is the complete integral when X = o. If y=Y-\-u (4) be substituted in equation (i), the result will be the sum of the results of putting y = Fand of putting y = u ; the first of these results will be X, because Fis an integral of equation (i), and the second will be zero because u is an integral of equa- tion (2). Hence equation (4) expresses an integral of (i); and since it contains the n arbitrary constants of equation (3), it is the complete integral of equation (i). With reference to this equation F is called " the particular integral," and u is called "the complementary function." The particular integral contains no arbitrary constant, and any two particular integrals may differ by any multiple of a term belonging to the comple- mentary function. If one term of the complementary function of a linear equation of the second order be known, the complete solution can be found. For let y l be the known term ; then, if y = yp be substituted in the first member, the coefficient of v in the result will be the same as if v were a constant : it will there- fore be zero, and v being absent, the result will be a linear equa- tion of the first order for v', the first derivative of v. Under 36 DIFFERENTIAL EQUATIONS. the same circumstances the order of any linear equation can in like manner be reduced by unity. A very simple relation exists between the coefficients of an exact linear equation. Taking, for example, the equation of the second order, and indicating derivatives by accents, if is exact, the first term of the integral will be P^y' Subtracting the derivative of this from the first member, the remainder is (/>, /V)y + P,y. The second term of the integral must therefore be (P l P ')y ; subtracting the derivative of this ex- pression, the remainder, (P t />/ -j- P<>"}y, must vanish. Hence P 9 PI -j- P " = o is the criterion for the exactness of the given equation. A similar result obviously extends to equa- tions of higher orders. V Prob. 61. Solve x (3 + x) -\- $y = o, noticing that e* is an integral. Ans. y c^ + c a (x a + 3^ -f 6.v + 6. v Prob. 62. Solve (x* x)-^-. -4- z(2x 4- i)-f- + 2y = o. dx ax Ans. (^ *) b y = f i(- a;4 ~~ 6' ra 4~ 2Jf -J 4x 3 log ^) + <r a ^: 3 . d*y . O d*y a dv v Prob. 63. Solve^rn + cos "^z ~ 2 sin P-^ y cos c/ = sin 2#. ac/ f at/ Ans. y e- sin '^ ^ sin *(^ + ART. 16. LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS. The linear equation with constant coefficients and second member zero may be written in the form J*-*y + . . . + A n y = o, (i) 7 Jt in which D stands for the operator -j-, D* for ^-,, etc., so that D" indicates that the operator is to be applied n times. Then, since ZV* = me mx , D*e mx = m*e mx , etc., it is evident that if LINEAR EQUATIONS, CONSTANT COEFFICIENTS. 37 y e mx be substituted in equation (l), the result after rejecting the factor e* will be Aftf + A^"- 1 + . . . + A H = o. (2) Hence, if m satisfies equation (2), e mx is an integral of equation (i) ; and if m lt m^ . . . m n are n distinct roots of equation (2), the complete integral of equation (i) will be y = c^ x + /"* + . . . + c n e" n *. (3) For example, if the given equation is dy the equation to determine m is n? m 2 = o, of which the roots are m l = 2, m t = i; therefore the in- tegral is y = cS* + c,e-\ The general equation (i) may be written in the symbolic form f(D) .y = o, in which / denotes a rational integral func- tion. Then equation (2) is f(m) = o, and, just as this last equation is equivalent to (m m^(m m t ) . . . (m m n ) = o, (4) so the symbolic equation f(D) . y = o may be written (D - m t )(D - m,) ..."(/>- m n }y = o. (5) This form of the equation shows that it is satisfied by each of the quantities which satisfy the separate equations (D - m,}y = o, (D m^y = o...(D m n )y = o ; (6) that is to say, by the separate terms of the complete integral. If two of the roots of equation (2) are equal, say to m lt two of the equations (6) become identical, and to obtain the full number of integrals we must find two terms corresponding to the equation (D-m l )y = o; (7) in other words, the complete integral of this equation of which ^ = e n ^ is known to be one integral. For this purpose we 38 DIFFERENTIAL EQUATIONS. put, as explained in the preceding article, y =7,^. By differen- tiation, Dy = De m \ x v = e m ^ x (m.y -J- Dv] ; therefore (D m^f^v = e m *Dv. (8) In like manner we find (D m^e m ^ x v = e m i*D*v. (9) Thus equation (7) is transformed to D*v = o, of which the complete integral is v = c,x -{-c t ; hence that of equation (7) is y = ev(c l x+Ct). (io> These are therefore the two terms corresponding to the squared factor (D m,Y in f(D}y = o. It is evident that, in a similar manner, the three terms corresponding to a case of three equal roots can be shown to- be c m ^(c^ -\- CyX -(- c a ), and so on. The pair of terms corresponding to a pair of imaginary- roots, say m l = a -\- ifi, m t a i/3, take the imaginary form Separating the real and imaginary parts of &* and e-#*, and changing the constants, the expression becomes e ax (A cos fix-\-B sin fix}. (\ i) For a multiple' pair of imaginary roots the constants A and B must be replaced by polynomials as above shown in the case of real roots. When the second member of the equation with constant coefficients is a function of X, the particular integral can also be made to depend upon the solution of linear equations of the first order. In accordance with the symbolic notation introduced above, the solution of the equation JL_ ay = X , or (D - a}y = x (12) is denoted by y = (D a)~ l X, so that, solving equation (12), we have D^=-a X = r/-T-r (13) as the value of the inverse symbol whose meaning is " that LINEAR EQUATIONS, CONSTANT COEFFICIENTS. - 39 function of x which is converted to X by the direct operation expressed by the symbol D a" Taking the most convenient special value of the indefinite integral in equation (13), it gives the particular integral of equation (12). In like manner, the par- ticular integral of f(D)y = X is denoted by the inverse symbol fi-fi-X. Now, with the notation employed above, the symbolic J\ i fraction may be decomposed into partial fractions with constant numerators thus: N in which each term is to be evaluated by equation (13), and may be regarded (by virtue of the constant involved in the indefinite integral) as containing one term of the complement- ary function. For example, the complete solution of the equation is thus found to be y = When X is a power of x the particular integral may be found as follows, more expeditiously than by the evaluation of the integrals in the general solution. For example, if X = x* the particular integral in this example may be evaluated by development of the inverse symbol, thus : _ i _ _ _! y ~ D*-D~2 X ~ 2 * The validity of this equation depends upon the fact that the operations expressed in the second member of f(D) = (D - mi )(D _,) + ...+(/>_,,) are commutative, hence the process of verification is the same as if the equation were an algebraic identity. This general solution was published by Boole in the Cambridge Math. Journal, First Series, vol. n, p. 114. It had, however, been previously published by Lobatto, Theorie des Characteristiques, Amster- dam, 1837. 40 DIFFERENTIAL EQUATIONS. The form of the operand shows that, in this case, it is only necessary to carry the development as far as the term contain- ing D\ For other symbolic methods applicable to special forms of X we must refer to the standard treatises on this subject. d*y dy Prob. 64. Solve JH-< 3-7 + y . ^dx dx Ans. y = <**(Ax + ) + c<r*. ' Prob. 65. Show that and that ~ sin (ax + ^) = T sin (ax + ft). Prob. 66. Solve (Z> 5 + i )y = e* + sin zx + sin x. (Compare Prob. 59, Art. 14.) Ans. y = A sin x -f- B cos x -\- %e* -J sin zx $x cos x. ART. 17. HOMOGENEOUS LINEAR EQUATIONS. The linear differential equation in which A , A^ etc., are constants, is called the "homogene- ous linear equation." It bears the same relation to x m that the equation with constant coefficients does to e mx . Thus, if yx m be substituted in this equation, the factor x m will divide out from the result, giving an equation for determining m, and the n roots of this equation will in general determine the n terms of the complete integral. For example, if in the equation jd*y . dy *'-4 + 2*/- - 2y - o da? dx we put y = x m , the result is m(m i) + 2m 2 o, or (m i)(m -}- 2) = o. , ^ The roots of this equation are m 1 = I and m 9 = 2. Hence y = c^x -j- CyX~* is the complete integral. Equation (i) might in fact have been reduced to the form with constant coefficients by changing the independent vari- HOMOGENEOUS LINEAR EQUATIONS. 41 able to 0, where x = e , or 6 = log x. We may therefore at once infer from the results established in the preceding article that the terms corresponding to a pair of equal roots are of the form (c l + c t log x)x m , (2) and also that the terms corresponding to a pair of imaginary roots, a ifi, are x*[A cos (/3 log x) + B sin (ft log *)]. (3) The analogy between the two classes of linear equations considered in this and the preceding article is more clearly seen when a single symbol $= xD is used for the operation of taking the derivative and then multiplying by x, so that $x m = mx m . It is to be noticed that the operation x^D 1 is not the same as & or xDxD, because the operations of taking the derivative and multiplying by a variable are not "commu- tative," that is, their order is not indifferent. We have, on the contrary, x^D* = 8(8 i) ; then the equation given above, which is (x*D* + 2xD 2)y = o, becomes [8(8 i) + 28 2]j = o, or (8-1X8 + 2)^ = 0, the function of 8 produced being the same as the function of m which is equated to o in finding the values of m. A linear equation of which the first member is homoge- neous and the second member a function of x may be reduced to the form A$).y = x- (4) The particular integral may, as in the preceding article (see eq. (14)), be separated into parts each of which depends upon the solution of a linear equation of the first order. Thus, solving the equation -ay = X, or (8 - a)y = X, (5) we find X=x" Cx- a - l Xdx. (6) a v 8 The more expeditious method which may be employed DIFFERENTIAL EQUATIONS. when X is a power of x is illustrated in the following example : d, v /z"i/ Given x* 2-f- = *'. The first member becomes homo- dx ax geneous when multiplied by x, and the reduced equation is ( 8 _ 3$'jjy X \ The roots of /($) =o are 3 and the double root zero, hence the complementary function is cj? -f- c ., -f- c 3 log x. Since in general f($)x r f(r)x r , we infer that in operating upon x* we may put $ = 3. This gives for the particular integral i i ^ _ i i ^ but fails with respect to the factor $ 3.* We therefore now fall back upon equation (6), which gives JT- x 3 = x* / x~ l dx = x* log x. The complete integral therefore is y = dy Prob. 67. Solve zx -~ 4- 3^-7 $y = x*. ax ax Ans. y = c 1 x + Prob. 68. Solve (*'>' + 3 ^Z>' + D)y = -. oc Ans. y = c, + <r, log x + <r,(log ^) a ART. 18. SOLUTIONS IN INFINITE SERIES. We proceed in this article to illustrate the method by which the integrals of a linear equation whose coefficients are algebraic functions of x may be developed in series whose terms are powers of x. For this purpose let us take the equation ' ' * The failure occurs because x 3 is a term of the complementary function having an indeterminate coefficient; accordingly the new term is of the same form as the second term necessary when 3 is a double root, but of course with a determinate coefficient. SOLUTIONS IN INFINITE SERIES. 43- which is known as " Bessel's Equation," and serves to define- the "Besselian Functions." If in the first member of this equation we substitute (or y the single term Ax m the result is A(m* - ri l }x m + Ax m +\ (2) the first term coming from the homogeneous terms of the equation and the second from the term x*y which is of higher degree. If this last term did not exist the equation would be satisfied by the assumed value of y, if m were determined so as to make the first term vanish, that is, in this case, by Ax n or Bx~ n . Now these are the first terms of two series each of which satisfies the equation. For, if we add to the value of y a term containing x m+2 , thus/ = A Q x m -j- A^" 1 ^ 2 , the new term- will give rise, in the result of substitution, to terms containing. x m+2 and x m+4 respectively, and it will be possible so to take A i that the entire coefficient of x m+ * shall vanish*. In like manner the proper determination of a third term makes the coefficient of x mJr * in the result of substitution vanish, and so on. We therefore at once assume = A,x' + A, x m + 2 -\- A t x m + 4 + . . . , (3) in which r has all integral values from o to oo. Substituting in equation (i) 2[{(m + 2/) 9 - n 9 }A^ m + 2r -\- ^X* +2(r+1) ] = o. (4) The coefficient of each power of x in this equation must sep- arately vanish ; hence, taking the coefficient of x m+2r , we have [(m + 2 ry-n>]A r +A r _ I =o. (5) When r = o, this reduces to m* n* = o, which determines the values of m, and for other values of r it gives ~ (m + 2r + n)(m -\-2r- n) Ar ~ 1 ' the relation between any two successive coefficients. For the first value of m, namely n, this relation becomes A __ !_ . A "'- ' -" 4-i DIFFERENTIAL EQUATIONS. whence, determining the successive coefficients in equation (3), the first integral of the equation is - -^ -, + wlw In like manner, the other integral is found to be r-jj - .. .J. (7) . . -, (8) and the complete integral is 7 = A y l -J- This example illustrates a special case which may arise in this form of solution. If n is a positive integer, the second series will contain infinite coefficients. For example, if n == 2, the third coefficient, or B v is infinite, unless we take B = o, in which case B^ is indeterminate and we have a repetition of the solution y r This will always occur when the same powers of x occur in the two series, including, of course, the case in which m has equal roots. For the mode of obtaining a new integral in such cases the complete treatises must be referred to.f It will be noticed that the simplicity of the relation between consecutive coefficients in this example is due to the fact that equation (i) contained but two groups of terms producing different powers of x, when Ax m is substituted for y as in ex- pression (2). The group containing the second derivative necessarily gives rise to a coefficient of the second degree in in, and from it we obtained two values of m. Moreover, be- cause the other group was of a degree higher by two units, the assumed series was an ascending one, proceeding by powers of x\ * The Besselian function of the wth order usually denoted byy is the value , of y\ above, divided by 2"! if n is a positive integer, or generally by 2 n r(n-\-i). For a complete discussion of these functions see Lommel's Studien liber die Bessel'schen Functionen, Leipzig, 1868; Todhunter's Treatise on Laplace's, Lame's and Bessel's Functions, London, 1875, etc - f A solution of the kind referred to contains as one term the product of the regular solution and log x, and is sometimes called a " logarithmic solution." See also American Journal of Mathematics, Vol. XI, p. 37. In the case of Bessel's equation, the logarithmic solution is the "Besselian Function of the iecond kind." SOLUTIONS IN INFINITE SERIES. 45 In the following example, there are also two such groups of terms, and their difference of degree shows that the series must ascend by simple powers. We assume therefore at once The result of substitution is - l ']= o. Equating to zero the coefficient of x" lJrr ~ 2 , (m -\- r + i)(m + r 2)A r + a(m + r i)A r . t = o, (12) which, when r = o, gives (;//+!)(* 2)/4.=o, (13) and when r > o, m-\-r I A r ^7 - i - i - \7 - i --- f^*r 1" (14) (in -\- r -{- i)(m -\-r~-2) The roots of equation (13) are m-=.2 and m = i; taking nt=2, the relation (14) becomes A i r+l A (r+*Y vh< nee the first integral is Taking the second value w = i, equation (14) gives r ~ 2 \- 3) , = gral is the finite expression whence B, = -- ^ , and ^, = o*; therefore the second inte~ * Bt would take the indeterminate form, and if we suppose it to have a finite value, the rest of the series is equivalent to B^y\, reproducing the first integral. 46 DIFFERENTIAL EQUATIONS. When the coefficient of the term of highest degree in the result of substitution, such as equation (11), contains m, it is possible to obtain a solution in descending powers of x. In this case, m occurring only in the first degree, but one such solution can be found; it would be identical with the finite integral (16). In the general case there will be two such solu- tions, and they will be convergent for values of x greater than unity, while the ascending series will converge for values less than unity.* When the second member of the equation is a power of x, the particular integral can be determined in the form of a series in a similar manner. For example, suppose the second mem- ber of equation (9) to have been x*. Then, making the sub- stitution as before, we have the same relation between consecu- tive coefficients; but when r = o, instead of equation (13) we have (m 4- i)(m 2)A x m ~ 2 = x to determine the initial term of the series. This gives m = 2$ and A -f ; hence, putting m = in equation (14), we find for the particular integral f 7 9.3 9.11.3.5 A linear equation remains linear for two important classes of transformations ; first, when the independent variable is changed to any function of x, and second, when for y we put vf(x). As an example of the latter, let y = e~ ax v be substituted in equation (9) above. After rejecting the factor e'**, the result is d*v dv 2v _ dx* dx x* Since this differs from the given equation only in the sign *When there are two groups of terms, the integrals are expressible in terms of Gauss's " Hypergeometric Series." f If the second member is a term of the complementary function (for ex- ample, in this case, if it is any integral power of x), the particular integral will take the logarithmic form referred to in the foot-note on p. 346. SYSTEMS OF DIFFERENTIAL EQUATIONS. 47 of a, we infer from equation (16) that it has the finite integral v = - -J . Hence the complete integral of equation (9) can X be written in the form xy <r,(2 ax) -f c^e- ax (2 -f- ax). <Ty Prob. 69. Integrate in series the equation -T-J + ^ry = o. Ans. ^(.-I^+Lj**-- . . .)-HB(_J^+'^'_ . . .). ^y . </y Prob. 70. Integrate in series x*. -, + x* + (x 2)y = o. Prob. 71. Derive for the equation of Prob. 70 the integral y 9 = e~*(x~ l + i + *)> an d find its relation to those found above. ART. 19. SYSTEMS OF DIFFERENTIAL EQUATIONS. It is shown in Art/ 12 that a determinate system of n differ- ential equations of the first order connecting n -J- I variables has for its complete solution as many integral equations con- necting the variables and also involving n constants of inte- gration. The result of eliminating n I variables would be a single relation between the remaining two variables containing in general the n constants. But the elimination may also be effected in the differential system, the result being in general an equation of the wth order of which the equation just men- tioned is the complete integral. For example, if there were two equations of the first order connecting the variables x and y with the independent variable /, by differentiating each we should have four equations from which to eliminate one vari- able, say y, and its two derivatives * with respect to /, leaving a single equation of the second order between x and /. It is easy to see that the same conclusions hold if some of the given equations are of higher order, except that the order of the result will be correspondingly higher, its index being in * In general, there would be n* equations from which to eliminate i variables and n derivatives of each, that is, (n i)( -f- i) = n* I quantities leaving a single equation of the wth order. 48 DIFFERENTIAL EQUATIONS. general the sum of the indices of the orders of the given equa- tions. The method is particularly applicable to linear equa- tions with constant coefficients, since we have a general method of solution for the final result. Using the symbolic notation, the differentiations are performed simply by multiplying by the symbol D, and therefore the whole elimination is of exactly the same form as if the equations were algebraic. For ex- ample, the system cFy dx _ dx dy when written symbolically, is (2D* 4)y Dx = 2t, whence, eliminating x, 2t 2 D*-4 D 2D 4D~ y ~ o which reduces to (D-i) Integrating, y = (A + Bty + Ce~* - \t, the particular integral being found by symbolic development, as explained at the end of Art. 16. The value of x found in like manner is x = (A' + B'ty + C'e-V - f The complementary function, depending solely upon the deter- minant of the first members,* is necessarily of the same form as that for y, but involves a new set of constants. The re- lations betv/een the constants is found by substituting the values of x and y in one of the given equations, and equating to zero in the resulting identity the coefficients of the several terms of the complementary function. In the present ex- ample we should thus find the value of x, in terms of A, B, and C, to be * The index of the degree in D of this determinant is that of the order of the final equation ; it is not necessarily the sum of the indices of the orders of the given equations, but cannot exceed this sum. SYSTEMS OF DIFFERENTIAL EQUATIONS. 49" In general, the solution of a system of differential equations depends upon our ability to combine them in such a way as to form exact equations. For example, from the dynamical system ~dJ- ' d?~ ' d? - where X, Y, Z are functions of x, y, and 2, but not of t t we form the equation dx ,dx . dy jdz . dz ,dz , , ,,, , - d -- r- -^-d-~ + -rd Xdx -4- Ydy -\-Zdz. dt dt '<#<#' dt dt The first member is an exact differential, and we know that for a conservative field of force the second member is also exact, that is, it is the differential of a function U of x> y, and z. The integral is that first integral of the system (i) which is known as the equation of energy for the unit mass. Just as in Art. 13 an equation of the second order was re- garded as equivalent to two equations of the first order, so the system (i) in connection with the equation defining the resolved velocities forms a system of six equations of the first order, of which system equation (2) is an " integral " in the sense ex- plained in Art. 12. Prob. 72. Solve the equations - = = dt as a system im- my mx ear in /. Ans. x A cosmt-\-smmt,y=A sinmtJB cosmt. Prob. 73. Solve the system -5 -- 1- *y = e*. -f- -4- z = o. dx dx Ans. y - Ae nx + Be~ nx + -r . z nAe nx -\- nEe'** -- ^ . n t n i Prob. 74. Find for the system - = x<t>(x,y), -^ = y<f>(x,y} a first integral independent of the function <f>. dy dx 50 DIFFERENTIAL EQUATIONS. Prob. 75. The approximate equations for the horizontal motion of a pendulum, when the earth's rotation is taken into account, are d*x dy , gx d*y . dx , gy 2f-> r i +i r = > i + * r is+i = < show that both x and y are of the form A cos ,/ -f- -B sin ,/ + C cos nj -f- D sin #, ART. 20. FIRST ORDER AND DEGREE WITH THREE VARIABLES. The equation of the first order and degree between three variables x, y and z may be written Pdx + Qdy + Rdz = o, (i) where P, Q and R are functions of x, y and z. When this equation is exact, P, Q and R are the partial derivatives of some function u, of x, y and z ; and we derive, as in Art. 4, 'dP = 3Q dQ = d_ -dR = -dP_ ,. -dy d*' dz ~ 'dy' d* dz for the conditions of exactness. In the case of two variables, when the equation is not exact integrating factors always exist; but in this case, there is not always a factor // such that j*P, pQ and pR (put in place of P, Q, and R) will satisfy all three of the conditions (2). It is easily shown that for this purpose the relation must exist between the given values of P, Q, and R. This is therefore the " condition of integrability " of equation (i).* When this condition is fulfilled equation (i) may be inte- grated by first supposing one variable, say z, to be constant. Thus, integrating Pdx -f- Qdy = o, and supposing the constant of integration C to be a function of z, we obtain the integral, so * When there are more than three variables such a condition of integra- bility exists for each group of three variables, but these conditions are not all independent. Thus with four variables there are but three independent con- ditions. FIRST ORDER AND DEGREE, THREE VARIABLES. 51 far as it depends upon x and y. Finally, by comparing the total differential of this result with the given equation we de- termine dC in terms of z and dz, and thence by integration the value of C. It may be noticed that when certain terms of an exact equation forms an exact differential, the remaining terms must also be exact. It follows that if one of the variables, say z can be completely separated from the other two (so that in equation (i) ^becomes a function of z only and P and Q func- tions of x and y, but not of z) the terms Pdx -f- Qdy must be thus rendered exact if the equation is integrable.* For example, zydx zxdy y*dz = o. is an integrable equation. Accordingly, dividing by y*2. which we notice separates the variable z from x and y, puts it in the exact form ydx xdy dz y of which the integral is x = y log cz. Regarding x, y and z as coordinates of a moving point, an integrable equation restricts the point to motion upon one of the surfaces belonging to the system of surfaces represented by the integral ; in other words, the point (x, y, z) moves in an arbitrary curve drawn on such a surface. Let us now consider in what way equation (i) restricts the motion of a point when it is not integrable. The direction cosines of a moving point are proportional to dx, dy, and dz; hence, denoting them by /, m and , the direction of motion of the point satisfying equation (i) must satisfy the condition Pl+Qm+Rn = o. (4) It is convenient to considernn this connection an auxiliary system of lines represented, as explained in Art. 12, by the simultaneous equations dx dy dz = = fe\ P Q R' *In fact for this case the condition (3) reduces to its last term, which ex- presses the exactness of Pdx -f- Qdy, 52 DIFFERENTIAL EQUATIONS. The direction cosines of a point moving in one of the lines of this system are proportional to P, Q and R. Hence, de- noting them by A, ju, v, equation (4) gives A/ -j- pin -(- vn = o (6) for the relation between the directions of two moving points, whose paths intersect, subject respectively to equation (i) and to equations (5). The paths in question therefore intersect at right angles; therefore equation (i) simply restricts a point to move in a path which cuts orthogonally the lines of the auxili- ary system. Now, if there be a system of surfaces which cut the auxiliary lines orthogonally, the restriction just mentioned is completely expressed by the requirement that the line shall lie on one of these surfaces, the line being otherwise entirely arbitrary.. This is the case in which equation (i) is integrable.* On the other hand, when the equation is not integrable, the restriction can only be expressed by two equations involving an arbitrary function. Thus if we assume in advance one such relation, we know from Art. 12 that the given equation (i) together with the first derivative of the assumed relation forms a system admitting of solution in the form of two integrals- Both of these integrals will involve the assumed. function. For any particular value of that function we have a system of lines satisfying equation (i), and the arbitrary character of the func- tion makes the solution sufficiently general to include all lines which satisfy the equation.f Prob. 76. Show that the equation (mz ny)dx + (nx lz]dy + (ly mx}dz = o is integrable, and infer from the integral the character of the auxil- *It follows that, with respect to the system of lines represented by equations (5), equation (3) is the condition that the system shall admit of surfaces cutting them orthogonally. The lines of force in any field of conservative forces form such a system, the orthogonal surfaces being the equipotential surfaces. f So too there is an arbitrary element about the path of a point when the single equation to which it is subject is integrable, but this enters only into one of the two equations necessary to define the path. PARTIAL EQUATIONS, FIRST ORDEX OO iary lines. (Compare the illustrative example at the end of Art. 12.) Ans. nx Iz = C(ny mz). Prob. 77. Solve yz'dx z*dy e*dz o. Ans. yz = e*(i-\-cz). Prob. 78. Find the equation which in connection with^ .f(x) forms the solution of dz = aydx -f- bdy. Prob. 79. Show that a general solution of ydx (x z)(dy dz) is given by the equations yz=(t>(x), y = (x z) (/>'(*). (This is an example of " Monge's Solution.") ART. 21. PARTIAL DIFFERENTIAL EQUATIONS OF FIRST ORDER AND DEGREE. Let x denote an unknown function of the two independent variables x and y, and let denote its partial derivatives : a relation between one or both of these derivatives and the variables is called a " partial dif- ferential equation " of the first order! A value of z in terms of x and y which with its derivatives satisfies the equation, or a relation between x, y and z which makes z implicitly such a function, is a " particular integral." The most general equation of this kind is called the " general integral." If only one of the derivatives, say/, occurs, the equation may be solved as an ordinary differential equation. For if y is considered as a constant,/ becomes the ordinary derivative of z with respect to x\ therefore, if in the complete integral of the equation thus regarded we replace the constant of integra- tion by an arbitrary function of j, we shall have a relation which includes all particular integrals and has the greatest pos- sible generality. It will be found that, in like manner, when both p and q are present, the general integral involves an arbi- trary function, We proceed to give Lagrange's solution of the equation of 54r DIFFERENTIAL EQUATIONS. the first order and degree, or " linear equation," which may be written in the form Pp+Qq = R, (I) P, Q and R denoting functions of x, y and z. Let u = a, in which u is a function of x, y and z, and #, a constant, be an integral of equation (i). Taking derivatives with respect to x and y respectively, we have and substitution of the values of / and q in equation (i) gives the symmetrical relation Consider now the system of simultaneous ordinary differ- ential equations dx_ _ dy__ dz_ ~J>~~Q^~R (3) Let u = a be one of the integrals (see Art. 12) of this sys- tem. Taking its total differential, "du 3 9 -^+-^ + --^=0: and since by equations (3) dx, dy and dz are proportional to />, <2 and 7?, we obtain by substitution which is identical with equation (2). It follows that every integral of the system (3) satisfies equation (i), and conversely, so that the general expression for the integrals of (3) will be the general integral of equation (i). Now let v == b be another integral of equations (3), so that v is also a function which satisfies equation (2). As explained in Art. 12, each of the equations u a, v = b is the equation of a surface passing through a singly infinite system of lines belonging to the doubly infinite system represented by equa- tions (3). What we require is the general expression for any PARTIAL EQUATIONS, FIRST ORDER. 55 surface passing through lines of the system (and intersecting none of them). It is evident that f(u, v) = f(a, b) = C is such an equation,* and accordingly f(u, v), where f is an arbitrary function, will be found to satisfy equation (2). Therefore, to solve equation (i), we find two independent integrals u = a, v = b of the auxiliary system (3), (sometimes called Lagrange's equations,) and then put = 00)> (4) an equation which is evidently equally general with/(#, v) = o. Conversely, it may be shown that any equation of the form (4), regarded as a primitive, gives rise to a definite partial differential equation of Lagrange's linear form. For, taking partial derivatives with respect to the independent variables x and j, we have 3*^3-3- and eliminating (f>'(v) from these equations, the term contain- ing/^ vanishes, giving the result 3 3; 37 P + (5) which is of the form Pp -f- Qq R.\ * Each line of the system is characterized by special values of a and b which we may call its coordinates, and the surface passes through those lines whose coordinates are connected by the perfectly arbitrary relation f(a, 6) = C. f These values of P, Q and R are known as the " Jacobians " of the pair of functions u, v with respect to the pairs of variables y, z ; z, x ; and x, y re- spectively. Owing to their analogy to the derivatives of a single function they are sometimes denoted thus : _ 3(, z/) 3(, 3(.v, g _ d(u, v) The Jacobian vanishes if the functions u and v are not independent, that is to say, if can be expressed identically as a function of v. In like manner, 56 DIFFERENTIAL EQUATIONS. As an illustration, let the given partial differential equa- tion be (mz ny]p -f- (nx lz)q = ly mx, (6) .for which Lagrange's Equations are dx dy dz _____^^__ * __ 17 > mz ny ~ nx Iz ~ ly mx' *'' These equations were solved at the end of Art. 12, the two integrals there found being Ix -j- my -\- nz = a and x* -\- y* -j- ^ = b*> (8) Hence in this case the system of " Lagrangean lines" con- sists of the entire system of circles having the straight line for axis. The general integral of equation (6) is then Ix -j- my + nz = (t>(x* -\- y* -f- z*), (10) which represents any surface passing through the circles just mentioned, that is, any surface of revolution of which (9) is the axis.* Lagrange's solution extends to the linear equation contain- ing n independent variables. Thus the equation being the auxiliary equations are dx\ dx^ _ _ dx n _ dz ~I\'-~-^\~- : ^ = ^~' ! - = o is the condition that (a function of x, y and 2) is expressible <H*, >, 2) identically as a function of u and v, that is to say, that = o shall be an in- tegral of Pp + Qq= R. * When the equation Pdx -f- Qdy + Rdz = o is integrable (as it is in the above example; see Prob. 76, Art. 20), its integral, which may be put in the form V = C, represents a singly infinite system of surfaces which the Lagrangean lines cut orthogonally ; therefore, in this case, the general integral may be de- fined as the general equation of the surfaces which cut orthogonally the system V = C. Conversely, starting with a given system V = C, u = J\v) is the gen- eral equation of the orthogonal surfaces, if u = a and v = b are integrals of COMPLETE AND GENERAL INTEGRALS. 5? and if u l = c lt u y = c a , . . . u n = c n are independent integrals, the most general solution is /(,, . . . ) = o, where /is an arbitrary function. ^ 2 ^ 2T / y\ Prob. 80. Solve xz- \-yz~~ = xy. Ans. xy z* =/[ ). dx dy \y) Prob. 81. Solve (y + z)p + (z + x}q = x + y. Prob. 82. Solve (x +^)(/ q) z. Ans. (x-\-y) log 2 x =/(x-\-y). Prob. 83. Solve x(y z)p -\-y(z x)q = z(x y). Ans. x -\- y-\- z= f(xyz). ART. 22. COMPLETE AND GENERAL INTEGRALS. We have seen in the preceding article that an equation be- tween three variables containing an arbitrary function gives rise to a partial differential equation of the linear form. It follows that, when the equation is not linear in / and q, the general integral cannot be expressed by a single equation of the f^rm 0(, v) = o; it will, however, still be f^und to- depend upon a single arbitrary function. X . It therefore becomes necessary to consider an integral hav- ing as much generality as can be given by the presence of arbi- trary constants. Such an equation is called a " complete in- tegral " ; it contains two arbitrary constants (n arbitrary con- stants in the general case of n independent variables), because this is the number which can be eliminated from such an equa- tion, considered as a primitive, and its two derived equations. For example, if (*-a)' + o/-)' + *' = /P, a and b being regarded as arbitrary, be taken as the primitive, the derived equations are x a -\- zp = o, y b -f- zq = O, and the elimination of a and b gives the differential equation A/ +V +!)*. of which therefore the given equation is a complete integral. 58 DIFFERENTIAL EQUATIONS. Geometrically, the complete integral represents a doubly in- finite system of surfaces ; in this case they are spherical sur- faces having a given radius and centers in the plane of xy. In general, a partial differential equation of the first order with two independent variables is of the form F(x, y, z, p, q) = O, (l) and a complete integral is of the form f(x, y, 2, a, b) = o. (2) In equation (i) suppose x, y and z to have special values, namely, the coordinates of a special point A ; the equation becomes a relation between p and q. Now consider any sur- face passing through A of which the equation is an integral of (i), or, as we may call it, a given "integral surface " passing through A. The tangent plane to this surface at A determines values of / and q which must satisfy the relation just men- tioned. Consider also those of the complete integral surfaces [equation (2)] which pass through A. They form a singly in- finite system whose-tangent planes at A have values of p and q which also satisfy the relation. There is obviously among them one which has the same value of /, and therefore also the same value of q, as the given integral. Thus there is one of the complete integral surfaces which touches at A the given integral surface. It follows that every integral surface (not in- cluded in the complete integral) must at every one of its points touch a surface included in the complete integral.* It is hence evident that every integral surface is the en- velope of a singly infinite system selected from the complete integral system. Thus, in the example at the beginning of this article, a right cylinder whose radius is k and whose axis lies in the plane of xy is an integral, because it is the envelope * Values of x, y, and z, determining a point, together with values of/ and q, determining the direction of a surface at that point, are said to constitute an "element of surface." The theorem shows that the complete integral is ' com- plete " in the sense of including all the surface elements which satisfy the differ- ential equation. The method of grouping the "consecutive" elements to form an integral surface is to a certain extent arbitrary. COMPLETE AND GENERAL INTEGRALS. 59 of those among the spheres represented by the complete in- tegral whose centers are on the axis of the cylinder. If we make the center of the sphere describe an arbitrary curve in the plane of xy we shall have the general integral in this ex- ample. In general, if in equation (2) an arbitrary relation between a and b, such as b = </>(a), be established, the envelope of the singly infinite system of surfaces thus defined will represent the general integral. By the usual process, the equation of the envelope is the result of eliminating a between the two equations o f(x, y, z, a, 0(rt) ) = o, ~Tf( x i y> z > a > 0(*) ) = - (3) These two equations together determine a line, namely, the " ultimate intersection of two consecutive surfaces." Such lines are called the " characteristics " of the differential equa- tion. They are independent of any particular form of the complete integral, being in fact lines along which all integral surfaces which pass through them touch one another. In the illustrative example above they are equal circles with centers in the plane of xy and planes perpendicular to it.* The example also furnishes an instance of a " singular so- lution " analogous to those of ordinary differential equations. *The characteristics are to be regarded not merely as lines, but as " linear elements of surface," since they determine at each of their points the direction of the surfaces passing through them. Thus, in the illustration, they are cir- cles regarded as great-circle elements of a sphere, or as elements of a right cylinder, and may be likened to narrow hoops. They constitute in all cases a triply infinite system. The surfaces of a complete integral system contain them all, but they are differently grouped in different integral surfaces. If we arbitrarily select a curve in space there will in general be at each of its points but one characteristic through which the selected curve passes; that is, whose tangent plane contains the tangent to the selected curve. These char- acteristics (for all points of the curve) form an integral surface passing through the selected curve ; and it is the only one which passes through it unless it be itself a characteristic. Integral surfaces of a special kind result when the se- lected curve is reduced to a point. In the illustration these are the results of rotating the circle about a line parallel to the axis of z. 60 DIFFERENTIAL EQUATIONS. For the planes z = k envelop the whole system of spheres represented by the complete integral, and indeed all the sur- faces included in the general integral. When a singular solu- tion exists it is included in the result of eliminating a and b from equation (2) and its derivatives with respect to a and b, that is, from but, as in the case of ordinary equations, this result may in- clude relations which are not solutions. Prob. 84. Derive a differential equation from the primitive 2x + my -f- nz = a, where /, m, n are connected by the relation f + m* + *' = i. Prob. 85. Show that the singular solution of the equation found in Prob. 84 represents a sphere, that the characteristics con- sist of all the straight lines which touch this sphere, and that the general integral therefore represents all developable surfaces which touch the sphere. Prob. 86. Find the integral which results from taking in the general integral above /' -\-m* = cos" (a constant) for the arbitrary relation between the parameters. ART, 23. COMPLETE INTEGRAL FOR SPECIAL FORMS. A complete integral of the partial differential equation F(x,y,z,p,q) = o (i) contains two constants, a and b. If a be regarded as fixed and b as an arbitrary parameter, it is the equation of a singly in- finite system of surfaces, of which one can be found passing through any given point. The ordinary differential equation of this system, which will be independent of b, may be put in the form dz = pdx -\- qdy, (2) in which the coefficients/ and q are functions of the variables and the constant a. Now the form of equation (2) shows that these quantities are the partial derivatives of z, in an integral of equation (i); therefore they are values of p and q which COMPLETE INTEGRAL FOR SPECIAL FORMS. 61 satisfy equation (i). Conversely, if values of/ and q in terms of the variables and a constant a which satisfy equation (i) are such as to make equation (2) the differential equation of a sys- tem of surfaces, these surfaces will be integrals. In other words, if we can find values of/ and q containing a constant a which satisfy equation (i) and make dz = pdx -f- qdy inte- grable, we can obtain by direct integration a complete inte- gral, the integration introducing a second constant. There are certain forms of equations for which such values of / and q are easily found. In particular there are forms in which / and q admit of constant values, and these obviously make equation (2) integrable. Thus, if the equation contains / and q only, being of the form F(P* 4) = 0, (3) we may put p = a and q = b, provided F(a,t)=o. (4) Equation (2) thus becomes dz = adx -f- bdy, whence we have the complete integral z = ax+by-\ r c, (5) in which a and b are connected by the relation (4) so that a, b and c are equivalent to two arbitrary constants. In the next place, if the equation is of the form *=P* + W+f(P,q\ (6) which is analogous to Clairaut's form, Art. 10, constant values of p and q are again admissible if they satisfy z = ax + by+f(a,b\ (7) and this is itself the complete integral. For this equation is of the form z ax + by + c, and expresses in itself the rela- tions between the three constants. Problem 84 of the preced- ing article is an example of this form. In the third place, suppose the equation to be of the form F(z,p,q} = o, (g) 62 DIFFERENTIAL EQUATIONS. in which neither x nor y appears explicitly. If we assume q = ap, p will be a function of z determined from F(z, p, ap] o, say / = (f>(z), (9) Then dz = pdx -\- qdy = o becomes dz (f)(z)(dx -|- ady), which is integrable, giving the complete integral A fourth case is that in which, while z does not explicitly occur, it is possible to separate x and / from y and q, thus put- ting the equation in the form /.(*>/) =/,(* ?) (II) If we assume each member of this equation equal to a con- stant #, we may determine/ and q in the forms / = <&(>> a \ q = fad', a). (12) and dz = pdx + qdy takes an integrable form giving (13) It is frequently possible to reduce a given equation by trans- formation of the variables to one of the four forms considered in this article.* For example, the equation x*p* + y q* = z* may be written *The general method, due to Charpit, of finding a proper value olp consists of establishing, by means of the condition of integrability, a linear partial dif- ferential equation for/, of which we need only a particular integral. This may be any value of / taken from the auxiliary equations employed in Lagrange's process. See Boole, Differential Equations (London 1865), p. 336 ; also For- syth, Differential Equations (London 1885), p. 316, in which the auxiliary equa- tions are deduced in a more general and symmetrical form, involving both / and q. These equations are in fact the equations of the characteristics regarded as in the concluding note to the preceding article. Denoting the partial deriva- tives of F(x, y, z,p, q) by X, Y, Z, P, Q, they are dx _ dy dz dt> >/,/ ' ~ ~ IP - Q Pp+Qq ~ X+ty ~ Y+Zq' See Jordan's Cours d'Analyse (Paris, 1887), vol. in, p. 318 ; Johnson's Differ- ential Equations (New York, 1889), p. 300. Any relation involving one or both the quantities /and q, combined with f=o, will furnish proper values of/ PARTIAL EQUATIONS, SECOND ORDER. 63 whence, putting x' = log x, y' = log y, z' = log 2, it becomes /'* + ^' a == l > which is of the form F(p', q'} = o, equation (3). Hence the integral is given by equation (5) when a? -j- &* = i; it may therefore be written z' = x' cos a-\-y' sin a -f- c, and restoring x, y, and #, that of the given equation is z = cx co * a y* Q ". Prob. 87. Find a complete integral for/ 1 q* = i. Ans. 2 = .# sec or + j> tan a + . Prob. 88. Find the singular solution of z = px + gy +/? Ans. 2 = xy. Prob. 89. Solve by transformation q = zyp*. Ans. z = ax -\~ ofy* + ^ Prob. 90. Solve z(p*q*} = x y. Ans. 2? = (# + a)i + (y + )1 + b. Prob. 91. Show that the solution given for the form F(z,p, q ) o represents cylindrical surfaces, and that F(z, o, o) = o is a singular solution. Prob. 92. Deduce by the method quoted in the foot-note two complete integrals of pq = px -j- qy. f x Y Ans. 22 = f -f~ a y) + A an( i z== xy-\-y ^(x'' -}- a) -\- b. ART. 24. PARTIAL EQUATIONS OF SECOND ORDER. We have seen in the preceding articles that the general solution of a partial differential equation of the first order de- pends upon an arbitrary function ; although it is only when the equation is linear in / and q that it is expressible by a single equation. But in the case of higher orders no general account can be given of the nature of a solution. Moreover, when we consider the equations derivable from a primitive con- taining arbitrary functions, there is no correspondence between their number and the order of the equation. For example, if and q. Sometimes several such relations are readily found ; for example, for the equation z = pq we thus obtain the two complete integrals and 4* =/ 64- DIFFERENTIAL EQUATIONS. the primitive with two independent variables contains two ar- bitrary functions, it is not generally possible to eliminate them and their derivatives from the primitive and its two derived equations of the first and three of the second order. Instead of a primitive containing two arbitrary functions, let us take an equation of the first order containing a single arbitrary function. This may be put in the form u = $(v\ u and v now denoting known functions of x, y, z, p, and q. (f>'(v) may now be eliminated from the two derived equations as in Art. 21. Denoting the second derivatives of z by 9V 9** ,_9!f = a*" ~ a-ra/ ~ ay the result is found to be of the form Rr + Ss-\-Tt+ U(rt - /) = V, (i) in which R, S, T, U, and V are functions of x, y, z, p, and,^. With reference to the differential equation of the second order the equation u = </>(v) is called an " intermediate equation of the first order " : it is analogous to the first integral of an ordi- nary equation of the second order. It follows that an inter- mediate equation cannot exist unless the equation is of the form (i); moreover, there are two other conditions which must exist between the functions R, S, T, and U. In some simple cases an intermediate equation can be ob- tained by direct integration. Thus, if the equation contains derivatives with respect to one only of the variables, it may be treated as an ordinary differential equation of the second order, the constants being replaced by arbitrary functions of the other variable. Given, for example, the equation xr p = xy, which may be written xdp pdx = xy dx. This becomes exact with reference to x when divided by x*, and gives the intermediate equation A second integration (and change in the form of the arbitrary function) gives the general integral z = $yx* log x -f PARTIAL EQUATIONS, SECOND ORDER. 65 Again, the equation p-\-r-\-s=i is already exact, and gives the intermediate equation which is of Lagrange's form. The auxiliary equations* are dz dx = dy = - -, x - z -f <t>(y) of which the first gives x y = a, and eliminating x from the second, its integral is of the form z = a -}- 0( y] 4" e ~ y b. Hence, putting b = $(a), we have for the final integral in which a further change is made in the form of the arbitrary function 0. Prob. 93. Solve / q = e* + e y . Ans. z = ye y c* + <p Prob. 94. Solve r+p* = y\ Ans. z = log [^00 - e-**] + Prob. 95. Solve /(j /) = x. Ans. 2 = (x +y) logj> + 0(*) + #(* Prob. 96. Solve /^ qr = o. Ans. ^c = 0(.y) + ^( 2 )- Prob. 97. Show that Monge's equations (see foot-note) give for Prob. 96 the intermediate integral p = (p(z) and hence derive the solution. * In Monge's method (for which the reader must be referred to the complete treatises) of finding an intermediate integral of Rr + Ss + Tt = V when one exists, the auxiliary equations Rdy* - Sdy dx + Tdo* = o, Rdp dy + Tdq dx = Vdx dy are established. These, in connection with dz = pdx -f- qdy, form an incomplete system of ordinary differential equations, between the five variables x, y, z, p, and q. But if it is possible to obtain two integrals of the system in the form = a, v = b t u = 0(z>) will be the intermediate integral. The first of the auxiliary equations is a quadratic giving two values for the ratio dy.dx. If these are distinct, and an intermediate integral can be found, for each, the values of p and q determined from them will make dz = pdx -(- qdy jntegrable, and give the general integral at once. 6G DIFFERENTIAL EQUATIONS. Prob. 98. Derive by Monge's method for q*r 2pqs -\- p*t = o the intermediate integral/ = q <fi(z), and thence the general integral. Ans. y + x<f>(z) = ART. 25. LINEAR PARTIAL DIFFERENTIAL EQUATIONS. Equations which are linear with respect to the dependent variable and its partial derivatives may be treated by a method analogous to that employed in the case of ordinary differential equations. We shall consider only the case of two independ- ent variables x and y, and put D- D' - " a*' ~ a/ so that the higher derivatives are denoted by the symbols D*, DD', D rt , D 3 , etc. Supposing further that the coefficients are constants, the equation may be written in the form f(D, D')z =. F(x, y\ (i) in which f denotes an algebraic function, or polynomial, of which the degree corresponds to the order of the differential equation. Understanding by an " integral" of this equation an explicit value of z in terms of x and y, it is obvious, as in Art. 1 5, that the sum of a particular integral and the general integral of AD, D'}z = o (2) will constitute an equally general solution of equation (i). It is, however, only when f(D, D') is a homogeneous function of D and D' that we can obtain a solution of equation (2) containing n arbitrary functions,* which solution is also the "comple- mentary function" for equation (i). Suppose then the equation to be of the form . and let us assume 2 = <p(y -f- mx), (4) * It is assumed that such a solution constitutes the general integral of an equation of the th order; for a primitive containing more than independent arbitrary functions cannot give rise by their elimination to an equation of the th order. LINEAR PARTIAL EQUATIONS. 67 where m is a constant to be determined. From equation (4), Dz = m(p'(y -f- mx) and D 'z = <j>\y -j- mx}, so that Dz = mD'z, D*z = m*D'*z, DD'z = mD'*y, etc. Substituting in equation (3) and rejecting the factor D' n z or (f> (n) (y -\- mx), we have Ajn* + A,m n - 1 -f . . . + A n = o (5) for the determination of m. If ;,, t 9 , . . . m n are distinct roots of this equation, * = <t>i(y + m **) +<P*( y + m **} + + <i>n(y + m n x) (6) is the general integral of equation (3). For example, the general integral of ; --., = o is thus QX oy found to be z = (f)(y -\- x] -j- $(y *) Any expression of the form Axy + Bx -f- Cy -\- D is a particular integral ; accordingly it is expressible as the sum of certain functions of x -\- y and x y respectively. The homogeneous equation (3) may now be written sym- bolically in the form (D - mjy)(D - m,D'} ...(D- m n D'}z = o, (7) in which the several factors correspond to the several terms of the general integral. If- two of the roots of equation (5) are equal, say, to m lt the corresponding terms in equation (6) are equivalent to a single arbitrary function. To form the general integral we need an integral of (D - m.DJz - o (8) in addition to 0(j -)- m } x}. This will in fact be the solution of (D - m,D'}z = <f>(y + ,*); (9) for, if we operate with D m^D' upon both members of this equation, we obtain equation (8). Writing equation (9) in the form p m,q = (f>(y + mx\ Lagrange's equations are -V . f . k j 1 0(7 + *!*) giving the integrals y -f- #*,.* = #, ^ = x<p(a) -}- b. Hence the integral of equation (9) is z = x(f)(y -|- m^x) -\- if)(y -\- ntji), (10) 63 DIFFERENTIAL EQUATIONS. and regarding also as arbitrary, these are the two independ- ent terms corresponding to the pair of equal roots. If equation (5) has a pair of imaginary roots m = ^ zV, the corresponding terms of the integral take the form which when and ip are real functions contain imaginary terms. If we restrict ourselves to real integrals we cannot now say that there are two radically distinct classes of inte- grals ; but if any real function of y -(- JJLX ~\- ivx be put in the form X-\-iY, either of the real functions X or Y will be an integral of the equation. Given, for example, the equation of which the general integral is s = 0(7 + "0 to obtain a real integral take either the real or the coefficient of the imaginary part of any real form of (f>(y -j- ix]. Thus, if 0(/) = # we find e y cosx and e y sin;r, each of which is an integral (see Chap. VI, p. 245). As in the corresponding case of ordinary equations, the particular integral of equation (i) may be made to depend upon the solution of linear equations of the first order. The inverse symbol -=? ^^(x, y) in the equation corresponding to equation (14), Art. 16, denotes the value of z in (D mD'}z F(x, y) or p mq = F(x, y). (l 1} For this equation Lagrange's auxiliary equations give y -j- mx = a, z = / F(x, a mx)dx -f- b = F^(x, a) -f- b, and the general integral is mx). (12) The first term, which is the particular integral, may there- fore be found by subtracting mx from y in F(x, y}, inte- LINEAR PARTIAL EQUATIONS. 69 grating with respect to x, and then adding mx to y in the result.* For certain forms of F(x, y) there exist more expeditious methods, of which we shall here only notice that which applies to the form F(ax -\- by). Since DF(ax -f- by) = aF'(ax -f by) and D'F(ax -f- by) = bF'(ax -\- by), it is readily inferred that, when f(D, D') is a homogeneous function of the wth degree, f(D,D')F(ax + by) = f(a, b)F*\ax + by). (13) That is, if t ax -\- by, the operation of f(D, D') on F(t) is equivalent to multiplication by/(a,b) and taking the th de- rivative, the final result being still a function of /. It follows that, conversely, the operation of the inverse symbol upon a function of t is equivalent to dividing by f(a, b) and integrating n times. Thus, ff. . .fPWT. (H) ^ y When ax -(- by is a multiple of y -f- w,-^, where m^ is a root of equation (5), this method fails with respect to the correspond- ing symbolic factor, giving rise to an equation of the form (9), of which the solution is given in equation (10). Given, for ex- ample, the equation 6*2 d*Z 6*2 or (D D') (D + 2D')z = sin (x y) -f sin (x + y). The complementary function is <p(y + *") + $(y 2x). The part of the particular integral arising from sin (x y), in which a = i, b = i, is - - / / sin tdf = ~ sin (x y). That aris- * The symbolic form of this theorem is corresponding to equation (13), Art. 16. The symbol e^xD 1 here indicates the addition of mx to y in the operand. Accordingly, using the expanded form of the symbol, f\ 22 X2 ffttxIJ f ( lA ~~" 1 1 | fttX ** ' ~~i ~^~ ~T~ ) f\ V) "^ f\ V ^~ fttX\ 8y 2 ! <5y* the symbolic expression of Taylor's Theorem. 70 DIFFERENTIAL EQUATIONS. ing from sin (x -\-y) which is of the form of a term in the com- plementary function is ^ jr, cos (x + j), which by equa- o tion (10) is \x cos (x -\-y). Hence the general integral of the given equation is * = 0( y + *) + $(y - 2*}+% sin (x -y)-\x cos (* + j).. If in the equation f(D, D')z = o the symbol f(D, D'), though not homogeneous with respect to D and D', can be separated into factors, the integral is still the sum of those corresponding to the several symbolic factors. The integral of a factor of the first degree is found by Lagrange's process; thus that of (D mD' a)z = o (15) is z = ^"0( y -\- mx). (16) But in the general case it is not possible to express the solution in a form involving arbitrary functions. Let us, how- ever, assume *=<***+*, (17) where c, h, and k are constants. Since De hx+ky = he kx + k:r and >'<**+**= k<* x + ky , substitution in f'(D,D'}z = o gives cf(fi, k)e hx ~ lrky = o. Hence we have a solution of the form (17) whenever h and k satisfy the relation Ak>k} = o, (18) c being altogether arbitrary. It is obvious that we may also write the more general solution z=2ce** +F <*, (19) where k = F(h) is derived from equation (18), and c and h admit of an infinite variety of arbitrary values. Again, since the difference of any two terms of the form fkx + FWy w jth different values of h is included in expression (19), we infer that the derivative of this expression with respect to h is also an integral, and in like manner the second and higher derivatives are integrals. For example, if the equation is LINEAR PARTIAL EQUATIONS. 71 for which equation (18) is f? k o, we have classes of in- tegrals of the forms 27)], e** + k (x -f 2/^) s + 6y(x -f- In particular, putting // = o we obtain the algebraic integrals Cl x y c&? + 27), ,(*' + 6*y), etc. The solution of a linear partial differential equation with variable coefficients may sometimes be effected by a change of the independent variables as illustrated in some of the exam- ples below. Prob. 99. Show that if m l is a triple root the corresponding. terms of the integral are o?<t>(y + m t x) -\- x$(y-{- w,#) B u 9 ' 2 9>j5 9 ^ Prob. 100. Solve 2^-^ -- 3^^ -- 2^ a = o. d* dxdy dy Prob. ici. Solve ^^ + 2 ^ + - 3 = ^. Ans. 2 = 0(^) -f- $(x +y)+ xx(x +y) y log *. Prob. 102. Solve (D 1 + $DD' + 6D n )z = (y - 2 X y\ Ans. z = (f>(y 2 x) + ip(y 3^) + x log (y 2x). tfz &z $z Prob. 103. Solve ^-; ^-^- -f ^ -- 2 = 0. Prob. 104. Show that for an equation of the form (15) the solu- tion given by equation (19) is equivalent to equation (16). i tfz i fiz i d*z i dz . Prob. 105. Solve ^-5 -- 1^- = ^1 -- ;^~ by transpose x dx x dx y dy y 9 dy tion to the independent variables x* an Prob.,06. So,ve,'l INDEX. Auxiliary system of lines, 51, 55. Besselian functions, 44 note. Bessel's equation, 43. Boundary (of real solutions), 17. Characteristics, 59. Clairaut's equation, 22, 61. Complementary function, 35, 48. Complete integral, 2, 30, 57, 60. Condition of integrability, 50. Cusp-locus, 17, 20. Decomposable equations, 13. Differentiation, solution by, 20. Direct integration, 2. Discriminant, 16, 18. Doubly infinite systems, 26, 31. Envelope, 15. Equation of energy, 29, 49. Equipotential surfaces, 52 note. Exact differentials, 2, 51. Exact equations, 6, 27, 30, 36. Extension of the linear equation, 12. Finite solutions, 45 First integral, 30, 31. First order and degree, i, 50. First order and second degree, 12. General integral, 53, 57. Geometrical applications, 23, 31. Geometrical representation of a differ ential equation, 3, 13, 15, 26. Homogeneous equations, 9. Homogeneous linear equations, 40 Integral, 26, 34, 49, 66. Integral equation, 26. Integral surface, 58. Integrating factors, 8, n, 33. Jacobians, 55 note. Lagrange's lines, 55. Lagrange's solution, 53, 56. Linear elements of surface, 59 note. Linear equations, 10, 34, 36. Linear partial differential equation, 66. Logarithmic solutions, 44 note, 46 note. Monge's method, 65, Prob. 97 note. Monge's solution, 53, Prob. 79 Node-locus, 19. Non-integrable equation, 51. Operative symbols, 36, 41. Order, equations of first, i. of second, 28. Orthogonal surfaces, 52 note, Orthogonal trajectories, 24. Parameters or arbitrary constants, 4, 15, 26, 31, 60. Partial differential equations. first order and degree, 53. linear, 66. / second order, 63 Particular integral. 2 determined in series, 46. of linear equation, 35, 38. 41 Pencil of curves, 14. Primitive 5 55 Separation of variables. 2, 51. Series, solutions in, 42, Simultaneous equations, 25, 47 Singular solutions, 15, 18, 26 note, 59. Symbolic solutions, 37 et seq., 41, 67. Systems of curves, 4, 26, 31. Systems of differential equations, 47. Tac locus, 1 6 Trajectories, 23 Transcendental and algebraic forms of solution, 2. Transformation of linear equations, 46. Ultimate intersections, 19, 59. q t3 <j* c t-P H n $ rH O .* UNIVERSITY OF TORONTO LIBRARY Do not re move the card from this Pocket. Acme Library Card Pocket Under Pat. " Ref. Index File." Made by LIBEAEY BUREAU II